Trigonometric Ratios and Functions

Trigonometric Ratios and Functions
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Trigonometric Ratios
and Functions
Right Triangle Trigonometry
Angles and Radian Measure
Trigonometric Functions of Any Angle
Graphing Sine and Cosine Functions
Graphing Other Trigonometric Functions
Modeling with Trigonometric Functions
Using Trigonometric Identities
Using Sum and Difference Formulas
Sundial (p. 518)
Tuning Fork (p.
(p 510)
Ferris Wheel (p.
(p 494)
SEE the Big Idea
Terminator (p. 476)
Parasail
iliing ((p.
p. 46
5))
Parasailing
465)
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Maintaining Mathematical Proficiency
Absolute Value
2
Example 1 Order the expressions by value from least to greatest: ∣ 6 ∣, ∣ −3 ∣, —, ∣ 10 − 6 ∣
∣ −4 ∣
∣6∣ = 6
2
∣ −3 ∣ = 3
1
2
2
4
∣ 10 − 6 ∣ = ∣ 4 ∣ = 4
—=—=—
∣ −4 ∣
The absolute value of a
negative number is positive.
2
So, the order is —, ∣ −3 ∣, ∣ 10 − 6 ∣, and ∣ 6 ∣.
∣ −4 ∣
Order the expressions by value from least to greatest.
1.
∣ 4 ∣, ∣ 2 − 9 ∣, ∣ 6 + 4 ∣, −∣ 7 ∣
3.
∣
−83
∣ −5 ∣
2. ∣ 9 − 3 ∣, ∣ 0 ∣, ∣ −4 ∣, —
⋅
∣, ∣ −2 8 ∣, ∣ 9 − 1 ∣, ∣ 9 ∣ + ∣ −2 ∣ − ∣ 1 ∣
4.
∣2∣
∣ −4 + 20 ∣, −∣ ∣, ∣ 5 ∣ − ∣ 3 ⋅ 2 ∣, ∣ −15 ∣
42
Pythagorean Theorem
Example 2 Find the missing side length of the triangle.
10 cm
26 cm
b
a2 + b2 = c2
Write the Pythagorean Theorem.
102 + b2 = 262
Substitute 10 for a and 26 for c.
100 +
Evaluate powers.
b2
= 676
b2 = 576
Subtract 100 from each side.
b = 24
Take positive square root of each side.
So, the length is 24 centimeters.
Find the missing side length of the triangle.
5.
6.
c
12 m
7.
b
9.6 mm
7 ft
25 ft
c
5m
9.
8.
a
10.
35 km
12
1
in.
3
7.2 mm
3
yd
10
b
a
1
yd
2
21 km
4 in.
11. ABSTRACT REASONING The line segments connecting the points (x1, y1), (x2, y1), and (x2, y2)
form a triangle. Is the triangle a right triangle? Justify your answer.
Dynamic Solutions available at BigIdeasMath.com
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45
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Mathematical
Practices
Mathematically proficient students reason quantitatively by creating
valid representations of problems.
Reasoning Abstractly and Quantitatively
Core Concept
The Unit Circle
y
The unit circle is a circle in the coordinate plane. Its center
is at the origin, and it has a radius of 1 unit. The equation of
the unit circle is
x2 + y2 = 1.
(0, 1)
(x, y)
θ
(−1, 0)
Equation of unit circle
(1, 0)
x
(0, 0)
As the point (x, y) starts at (1, 0) and moves counterclockwise
around the unit circle, the angle θ (the Greek letter theta) moves
from 0° through 360°.
(0, −1)
Finding Coordinates of a Point on the Unit Circle
Find the exact coordinates of the point (x, y) on the unit circle.
y
SOLUTION
(0, 1)
(x, y)
Because θ = 45°, (x, y) lies on the line y = x.
x2 + y2 = 1
Write equation of unit circle.
x2 + x2 = 1
Substitute x for y.
2x2 = 1
(−1, 0)
45°
x
(0, 0)
Add like terms.
1
x2 = —
2
1
x=—
—
√2
(1, 0)
(0, −1)
Divide each side by 2.
Take positive square root of each side.
(
1
1
) (
—
—
√2 √2
2 2
)
The coordinates of (x, y) are —
—, —
— , or —, — .
√2 √2
Monitoring Progress
Find the exact coordinates of the point (x, y) on the unit circle.
1.
2.
y
3.
y
(0, 1)
y
(0, 1)
(0, 1)
(x, y)
135°
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
(0, 0)
(0, 0)
(−1, 0)
225°
(1, 0)
(0, 0)
x
x
315°
(x, y)
(0, −1)
460
Chapter 9
hsnb_alg2_pe_09op.indd 460
(0, −1)
(x, y)
(0, −1)
Trigonometric Ratios and Functions
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Mathematical
Practices
Mathematically proficient students reason quantitatively by creating
a coherent representation of the problem at hand. (MP2)
Reasoning Abstractly and Quantitatively
Core Concept
The Unit Circle
y
The unit circle is a circle in the coordinate plane. Its center
is at the origin, and it has a radius of 1 unit. The equation of
the unit circle is
x2 + y2 = 1.
(0, 1)
(x, y)
θ
(−1, 0)
Equation of unit circle
(1, 0)
x
(0, 0)
As the point (x, y) starts at (1, 0) and moves counterclockwise
around the unit circle, the angle θ (the Greek letter theta) moves
from 0° through 360°.
(0, −1)
Finding Coordinates of a Point on the Unit Circle
Find the exact coordinates of the point (x, y) on the unit circle.
SOLUTION
Because θ = 45°, (x, y) lies on the line y = x.
y
x2 + y2 = 1
Write equation of unit circle.
x2 + x2 = 1
Substitute x for y.
2x2 = 1
(0, 1)
(x, y)
(−1, 0)
Add like terms.
45°
(1, 0)
x
(0, 0)
1
x2 = —
2
1
x=—
—
√2
Divide each side by 2.
Take positive square root of each side.
(
) (
—
—
(0, −1)
)
√2 √2
The coordinates of (x, y) are —
—, —
— , or —, — .
2 2
√2 √2
1
1
Monitoring Progress
Find the exact coordinates of the point (x, y) on the unit circle.
1.
2.
y
3.
y
(0, 1)
y
(0, 1)
(0, 1)
(x, y)
135°
(−1, 0)
(1, 0)
(−1, 0)
(1, 0)
x
(0, 0)
(0, 0)
(−1, 0)
225°
(1, 0)
(0, 0)
x
315°
(x, y)
(0, −1)
460
Chapter 9
Trigonometric Ratios and Functions
(0, −1)
(x, y)
(0, −1)
x
9.1
Right Triangle Trigonometry
Essential Question
How can you find a trigonometric function of
opp.
Tangent tan θ = —
adj.
adj.
Cotangent cot θ = —
opp.
hyp.
sec θ = —
adj.
hyp.
csc θ = —
opp.
Secant
te
nu
adj.
cos θ = —
hyp.
Cosine
Cosecant
po
opp.
sin θ = —
hyp.
hy
Sine
se
Consider one of the acute angles θ of a right triangle.
Ratios of a right triangle’s side lengths are used to
define the six trigonometric functions, as shown.
opposite side
an acute angle θ?
θ
adjacent side
Trigonometric Functions of Special Angles
Work with a partner. Find the exact values of the sine, cosine, and tangent functions
for the angles 30°, 45°, and 60° in the right triangles shown.
60°
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in
math, you need to
understand and use stated
assumptions, definitions,
and previously established
results in constructing
arguments.
2
45°
2
1
30°
1
45°
1
3
Exploring Trigonometric Identities
Work with a partner.
Use the definitions of the trigonometric functions to explain why each trigonometric
identity is true.
a. sin θ = cos(90° − θ)
b. cos θ = sin(90° − θ)
1
c. sin θ = —
csc θ
1
d. tan θ = —
cot θ
Use the definitions of the trigonometric functions to complete each trigonometric
identity.
e. (sin θ)2 + (cos θ)2 =
f. (sec θ)2 − (tan θ)2 =
Communicate Your Answer
3. How can you find a trigonometric function of an acute angle θ?
4. Use a calculator to find the lengths x and y
of the legs of the right triangle shown.
1
y
25°
x
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Lesson
9.1
What You Will Learn
Evaluate trigonometric functions of acute angles.
Find unknown side lengths and angle measures of right triangles.
Core Vocabul
Vocabulary
larry
Use trigonometric functions to solve real-life problems.
sine, p. 462
cosine, p. 462
tangent, p. 462
cosecant, p. 462
secant, p. 462
cotangent, p. 462
The Six Trigonometric Functions
Consider a right triangle that has an acute angle θ
(the Greek letter theta). The three sides of the triangle
are the hypotenuse, the side opposite θ, and the side
adjacent to θ.
Previous
right triangle
hypotenuse
acute angle
Pythagorean Theorem
reciprocal
complementary angles
hypotenuse
opposite
side
Ratios of a right triangle’s side lengths are used to
define the six trigonometric functions: sine, cosine,
tangent, cosecant, secant, and cotangent. These six
functions are abbreviated sin, cos, tan, csc, sec, and
cot, respectively.
θ
adjacent side
Core Concept
Right Triangle Definitions of Trigonometric Functions
Let θ be an acute angle of a right triangle. The six trigonometric functions of θ are
defined as shown.
REMEMBER
The Pythagorean Theorem
states that a2 + b2 = c2
for a right triangle with
hypotenuse of length c
and legs of lengths a
and b.
a
c
b
opposite
sin θ = —
hypotenuse
adjacent
cos θ = —
hypotenuse
opposite
tan θ = —
adjacent
hypotenuse
csc θ = —
opposite
hypotenuse
sec θ = —
adjacent
adjacent
cot θ = —
opposite
The abbreviations opp., adj., and hyp. are often used to represent the side lengths
of the right triangle. Note that the ratios in the second row are reciprocals of the
ratios in the first row.
1
csc θ = —
sin θ
1
sec θ = —
cos θ
1
cot θ = —
tan θ
Evaluating Trigonometric Functions
Evaluate the six trigonometric functions of the angle θ.
5
SOLUTION
From the Pythagorean Theorem, the length of the
hypotenuse is
θ
hypotenuse
12
—
hyp. = √ 52 + 122
—
= √ 169
= 13.
Using adj. = 5, opp. = 12, and hyp. = 13, the values of the six trigonometric
functions of θ are:
462
Chapter 9
hsnb_alg2_pe_0901.indd 462
opp. 12
sin θ = — = —
hyp. 13
5
adj.
cos θ = — = —
hyp. 13
opp. 12
tan θ = — = —
adj.
5
hyp. 13
csc θ = — = —
opp. 12
hyp. 13
sec θ = — = —
adj.
5
5
adj.
cot θ = — = —
opp. 12
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Evaluating Trigonometric Functions
In a right triangle, θ is an acute angle and sin θ = —47 . Evaluate the other five
trigonometric functions of θ.
SOLUTION
Step 1 Draw a right triangle with acute angle θ such that
the leg opposite θ has length 4 and the hypotenuse
has length 7.
7
4
Step 2 Find the length of the adjacent side. By the
Pythagorean Theorem, the length of the other leg is
adj. =
θ
33
—
—
adj. = √ 72 − 42 = √ 33 .
Step 3 Find the values of the remaining five trigonometric functions.
4
hyp. 7
Because sin θ = —, csc θ = — = —. The other values are:
7
opp. 4
—
—
4√ 33
4
opp.
tan θ = — = —
— = —
adj.
33
√33
√33
adj.
cos θ = — = —
hyp.
7
—
7√ 33
7
hyp.
sec θ = — = —
— = —
adj.
33
√33
Monitoring Progress
—
√33
adj.
cot θ = — = —
opp.
4
Help in English and Spanish at BigIdeasMath.com
Evaluate the six trigonometric functions of the angle θ.
1.
2.
3
θ
3.
θ
17
5 2
θ
5
15
4
7
4. In a right triangle, θ is an acute angle and cos θ = —
. Evaluate the other five
10
trigonometric functions of θ.
The angles 30°, 45°, and 60° occur frequently in trigonometry. You can use the
trigonometric values for these angles to find unknown side lengths in special
right triangles.
Core Concept
Trigonometric Values for Special Angles
The table gives the values of the six trigonometric functions for the angles 30°,
45°, and 60°. You can obtain these values from the triangles shown.
θ
2 30°
45°
1
45° 1
—
45°
—
60°
√3
—
2
—
√2
2
—
Section 9.1
hsnb_alg2_pe_0901.indd 463
1
2
30°
3
2
60°
1
sin θ
cos θ
—
√3
2
—
tan θ
1
—
2
√3
3
2
1
√2
—
√3
sec θ
—
—
—
√2
2
—
csc θ
—
2√3
3
—
—
—
—
√3
√2
1
2
√3
—
3
—
2√3
—
3
cot θ
—
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Finding Side Lengths and Angle Measures
Finding an Unknown Side Length
Find the value of x for the right triangle.
8
SOLUTION
30°
x
Write an equation using a trigonometric function that
involves the ratio of x and 8. Solve the equation for x.
adj.
cos 30° = —
hyp.
—
√3 x
—=—
2
8
Write trigonometric equation.
Substitute.
—
4√ 3 = x
Multiply each side by 8.
—
The length of the side is x = 4√3 ≈ 6.93.
Finding all unknown side lengths and angle measures of a triangle is called solving
the triangle. Solving right triangles that have acute angles other than 30°, 45°, and 60°
may require the use of a calculator. Be sure the calculator is set in degree mode.
READING
Throughout this chapter,
a capital letter is used
to denote both an angle
of a triangle and its
measure. The same letter
in lowercase is used to
denote the length of the
side opposite that angle.
Using a Calculator to Solve a Right Triangle
Solve △ABC.
B
c
SOLUTION
Because the triangle is a right triangle, A and B are
complementary angles. So, B = 90° − 28° = 62°.
a
28°
b = 15
A
C
Next, write two equations using trigonometric functions, one that involves the ratio
of a and 15, and one that involves c and 15. Solve the first equation for a and the
second equation for c.
opp.
tan 28° = —
adj.
a
tan 28° = —
15
15(tan 28°) = a
7.98 ≈ a
Write trigonometric equation.
Substitute.
hyp.
sec 28° = —
adj.
c
sec 28° = —
15
1
15 — = c
cos 28°
(
Solve for the variable.
)
16.99 ≈ c
Use a calculator.
So, B = 62º, a ≈ 7.98, and c ≈ 16.99.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
5. Find the value of x for the right triangle shown.
6
45°
x
B
c
A
464
b
Chapter 9
hsnb_alg2_pe_0901.indd 464
a
C
Solve △ABC using the diagram at the left and the given measurements.
6. B = 45°, c = 5
7. A = 32°, b = 10
8. A = 71°, c = 20
9. B = 60°, a = 7
Trigonometric Ratios and Functions
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Solving Real-Life Problems
Using Indirect Measurement
FINDING AN
ENTRY POINT
The tangent function is
used to find the unknown
distance because it involves
the ratio of x and 2.
You are hiking near a canyon. While standing at A,
you measure an angle of 90º between B and C, as
shown. You then walk to B and measure an angle of
76° between A and C. The distance between A and B
is about 2 miles. How wide is the canyon between
A and C?
C
x
SOLUTION
x
tan 76° = —
2
2(tan 76°) = x
Write trigonometric equation.
B 76°
2 mi A
Multiply each side by 2.
8.0 ≈ x
Use a calculator.
The width is about 8.0 miles.
If you look at a point above you, such as the top of
a building, the angle that your line of sight makes
with a line parallel to the ground is called the angle
of elevation. At the top of the building, the angle
between a line parallel to the ground and your line
of sight is called the angle of depression. These
two angles have the same measure.
angle of
depression
angle of
elevation
you
Using an Angle of Elevation
A parasailer is attached to a boat with a rope 72 feet long. The angle of elevation from
the boat to the parasailer is 28°. Estimate the parasailer’s height above the boat.
SOLUTION
Step 1 Draw a diagram that represents the situation.
72 ft
28°
h
Step 2 Write and solve an equation to find the height h.
h
sin 28° = —
72
Write trigonometric equation.
72(sin 28°) = h
Multiply each side by 72.
33.8 ≈ h
Use a calculator.
The height of the parasailer above the boat is about 33.8 feet.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. In Example 5, find the distance between B and C.
11. WHAT IF? In Example 6, estimate the height of the parasailer above the boat
when the angle of elevation is 38°.
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Exercises
9.1
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE In a right triangle, the two trigonometric functions of θ that are defined
using the lengths of the hypotenuse and the side adjacent to θ are __________ and __________.
2. VOCABULARY Compare an angle of elevation to an angle of depression.
3. WRITING Explain what it means to solve a right triangle.
4. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
What is the cosecant of θ ?
1
What is —?
sin θ
6
4
What is the ratio of the side opposite θ to the hypotenuse?
θ
What is the ratio of the hypotenuse to the side opposite θ?
Monitoring Progress and Modeling with Mathematics
12. ANALYZING RELATIONSHIPS Evaluate the six
In Exercises 5–10, evaluate the six trigonometric
functions of the angle θ. (See Example 1.)
5.
6.
θ
θ
9
In Exercises 13–18, let θ be an acute angle of a right
triangle. Evaluate the other five trigonometric functions
of θ. (See Example 2.)
8
12
7.
9
θ
5
7
14. cos θ = —
12
5
7
16. csc θ = —
8
13. sin θ = —
11
6
8.
7
3
15
15. tan θ = —6
14
17. sec θ = —
9
θ
9.
trigonometric functions of the 90° − θ angle in
Exercises 5–10. Describe the relationships you notice.
16
18. cot θ = —
11
10.
10
θ
14
19. ERROR ANALYSIS Describe and correct the error in
finding sin θ of the triangle below.
θ
18
26
17
11. REASONING Let θ be an acute angle of a right
triangle. Use the two trigonometric functions
—
4
√97
tan θ = — and sec θ = — to sketch and label
9
9
the right triangle. Then evaluate the other four
trigonometric functions of θ.
466
Chapter 9
hsnb_alg2_pe_0901.indd 466
8
θ
15
✗
opp. 15
sin θ = — = —
hyp. 17
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20. ERROR ANALYSIS Describe and correct the error in
finding csc θ, given that θ is an acute angle of a right
7
triangle and cos θ = —
.
11
✗
1
11
csc θ = — = —
7
cos θ
41. MODELING WITH MATHEMATICS To measure the
width of a river, you plant a stake on one side of the
river, directly across from a boulder. You then walk
100 meters to the right of the stake and measure a
79° angle between the stake and the boulder. What is
the width w of the river? (See Example 5.)
Not drawn to scale
In Exercises 21–26, find the value of x for the right
triangle. (See Example 3.)
21.
23.
79°
9
6
60°
x
60°
x
100 m
24.
30°
42. MODELING WITH MATHEMATICS Katoomba Scenic
Railway in Australia is the steepest railway in the
world. The railway makes an angle of about 52° with
the ground. The railway extends horizontally about
458 feet. What is the height of the railway?
30°
12
13
x
25.
w
22.
43. MODELING WITH MATHEMATICS A person whose
x
26.
8
45°
7
x
45°
x
eye level is 1.5 meters above the ground is standing
75 meters from the base of the Jin Mao Building in
Shanghai, China. The person estimates the angle
of elevation to the top of the building is about 80°.
What is the approximate height of the building?
(See Example 6.)
44. MODELING WITH MATHEMATICS The Duquesne
USING TOOLS In Exercises 27–32, evaluate the
trigonometric function using a calculator. Round your
answer to four decimal places.
27. cos 14°
28. tan 31°
29. csc 59°
30. sin 23°
31. cot 6°
32. sec 11°
Incline in Pittsburgh, Pennsylvania, has an angle of
elevation of 30°. The track has a length of about
800 feet. Find the height of the incline.
45. MODELING WITH MATHEMATICS You are standing
on the Grand View Terrace viewing platform at Mount
Rushmore, 1000 feet from the base of the monument.
Not drawn to scale
In Exercises 33–40, solve △ABC using the diagram and
the given measurements. (See Example 4.)
1000 ft
A
b
C
c
a
B
33. B = 36°, a = 23
34. A = 27°, b = 9
35. A = 55°, a = 17
36. B = 16°, b = 14
37. A = 43°, b = 31
38. B = 31°, a = 23
39. B = 72°, c = 12.8
40. A = 64°, a = 7.4
a. You look up at the top of Mount Rushmore at an
angle of 24°. How high is the top of the monument
from where you are standing? Assume your eye
level is 5.5 feet above the platform.
b. The elevation of the Grand View Terrace is
5280 feet. Use your answer in part (a) to find the
elevation of the top of Mount Rushmore.
46. WRITING Write a real-life problem that can be solved
using a right triangle. Then solve your problem.
Section 9.1
hsnb_alg2_pe_0901.indd 467
b
24°
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47. MATHEMATICAL CONNECTIONS The Tropic of
Cancer is the circle of
latitude farthest north
Tropic of
of the equator where
Cancer
the Sun can appear
equator
directly overhead. It lies
23.5° north of the equator,
as shown.
50. PROBLEM SOLVING You measure the angle of
elevation from the ground to the top of a building as
32°. When you move 50 meters closer to the building,
the angle of elevation is 53°. What is the height of
the building?
North Pole
23.5°
South Pole
51. MAKING AN ARGUMENT Your friend claims it is
possible to draw a right triangle so the values of the
cosine function of the acute angles are equal. Is your
friend correct? Explain your reasoning.
a. Find the circumference of the Tropic of Cancer
using 3960 miles as the approximate radius
of Earth.
b. What is the distance between two points on the
Tropic of Cancer that lie directly across from
each other?
52. THOUGHT PROVOKING Consider a semicircle with a
radius of 1 unit, as shown below. Write the values of
the six trigonometric functions of the angle θ. Explain
your reasoning.
48. HOW DO YOU SEE IT? Use the figure to answer
each question.
θ
y
90° − θ
h
θ
53. CRITICAL THINKING A procedure for approximating
x
π based on the work of Archimedes is to inscribe a
regular hexagon in a circle.
a. Which side is adjacent to θ ?
b. Which side is opposite of θ ?
c. Does cos θ = sin(90° − θ)? Explain.
30°
1
x
49. PROBLEM SOLVING A passenger in an airplane sees
30°
1
two towns directly to the left of the plane.
a. Use the diagram to solve for x. What is the
perimeter of the hexagon?
15° 25°
25,000 ft
d
x
b. Show that a regular n-sided polygon inscribed
in a circle of radius 1 has a perimeter of
180 °
2n sin — .
n
y
⋅
a. What is the distance d from the airplane to the
first town?
( )
c. Use the result from part (b) to find an expression
in terms of n that approximates π. Then evaluate
the expression when n = 50.
b. What is the horizontal distance x from the airplane
to the first town?
c. What is the distance y between the two towns?
Explain the process you used to find your answer.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Perform the indicated conversion. (Skills Review Handbook)
54. 5 years to seconds
55. 12 pints to gallons
56. 5.6 meters to millimeters
Find the circumference and area of the circle with the given radius or diameter.
(Skills Review Handbook)
57. r = 6 centimeters
468
Chapter 9
hsnb_alg2_pe_0901.indd 468
58. r = 11 inches
59. d = 14 feet
Trigonometric Ratios and Functions
2/5/15 1:48 PM
9.2
Angles and Radian Measure
Essential Question
How can you find the measure of an angle
in radians?
Let the vertex of an angle be at the origin, with one side of the angle on the positive
x-axis. The radian measure of the angle is a measure of the intercepted arc length on
a circle of radius 1. To convert between degree and radian measure, use the fact that
π radians
180°
— = 1.
Writing Radian Measures of Angles
Work with a partner. Write the radian measure of each angle with the given
degree measure. Explain your reasoning.
a.
b.
y
y
90°
radian
measure
degree
measure
60°
120°
135°
45°
π
30°
150°
0°
360° x
180°
x
210°
225°
315°
330°
240°
270°
300°
Writing Degree Measures of Angles
Work with a partner. Write the degree measure of each angle with the given
radian measure. Explain your reasoning.
y
degree
measure
radian
measure
7π
9
5π
9
4π
9
2π
9
x
11π
9
13π 14π
9
9
REASONING
ABSTRACTLY
To be proficient in math,
you need to make sense
of quantities and their
relationships in problem
situations.
16π
9
Communicate Your Answer
3. How can you find the measure of an angle
in radians?
y
4. The figure shows an angle whose measure is
30 radians. What is the measure of the angle in
degrees? How many times greater is 30 radians
than 30 degrees? Justify your answers.
x
30 radians
Section 9.2
hsnb_alg2_pe_0902.indd 469
Angles and Radian Measure
469
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9.2 Lesson
What You Will Learn
Draw angles in standard position.
Find coterminal angles.
Core Vocabul
Vocabulary
larry
initial side, p. 470
terminal side, p. 470
standard position, p. 470
coterminal, p. 471
radian, p. 471
sector, p. 472
central angle, p. 472
Previous
radius of a circle
circumference of a circle
Use radian measure.
Drawing Angles in Standard Position
In this lesson, you will expand your study of angles to include angles with measures
that can be any real numbers.
Core Concept
Angles in Standard Position
90° y
terminal
side
In a coordinate plane, an angle can be formed
by fixing one ray, called the initial side, and
rotating the other ray, called the terminal side,
about the vertex.
0°
An angle is in standard position when its vertex
is at the origin and its initial side lies on the
positive x-axis.
x
180° vertex initial
360°
side
270°
The measure of an angle is positive when the rotation of its terminal side is
counterclockwise and negative when the rotation is clockwise. The terminal side
of an angle can rotate more than 360°.
Drawing Angles in Standard Position
Draw an angle with the given measure in standard position.
a. 240°
b. 500°
c. −50°
b. Because 500° is 140°
more than 360°, the
terminal side makes
one complete rotation
360° counterclockwise
plus 140° more.
c. Because −50° is
negative, the terminal
side is 50° clockwise
from the positive
x-axis.
SOLUTION
a. Because 240° is 60°
more than 180°, the
terminal side is 60°
counterclockwise past
the negative x-axis.
y
y
240°
y
140°
x
500°
60°
Monitoring Progress
x
x
−50°
Help in English and Spanish at BigIdeasMath.com
Draw an angle with the given measure in standard position.
1. 65°
470
Chapter 9
hsnb_alg2_pe_0902.indd 470
2. 300°
3. −120°
4. −450°
Trigonometric Ratios and Functions
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Finding Coterminal Angles
STUDY TIP
If two angles differ by a
multiple of 360°, then the
angles are coterminal.
In Example 1(b), the angles 500° and 140° are coterminal because their terminal
sides coincide. An angle coterminal with a given angle can be found by adding or
subtracting multiples of 360°.
Finding Coterminal Angles
Find one positive angle and one negative angle that are coterminal with (a) −45°
and (b) 395°.
SOLUTION
There are many such angles, depending on what multiple of 360° is added or
subtracted.
a. −45° + 360° = 315°
−45° − 360° = −405°
b. 395° − 360° = 35°
395° − 2(360°) = −325°
y
y
−325°
35°
315°
−45°
x
395°
−405°
Monitoring Progress
x
Help in English and Spanish at BigIdeasMath.com
Find one positive angle and one negative angle that are coterminal with the
given angle.
5. 80°
STUDY TIP
Notice that 1 radian
is approximately equal
to 57.3°.
180° = π radians
180°
π
— = 1 radian
57.3° ≈ 1 radian
6. 230°
8. −135°
7. 740°
Using Radian Measure
Angles can also be measured in radians. To define
a radian, consider a circle with radius r centered at
the origin, as shown. One radian is the measure of
an angle in standard position whose terminal side
intercepts an arc of length r.
y
r
1 radian
Because the circumference of a circle is 2πr, there
are 2π radians in a full circle. So, degree measure
and radian measure are related by the equation
360° = 2π radians, or 180° = π radians.
r
x
Core Concept
Converting Between Degrees and Radians
Degrees to radians
Radians to degrees
Multiply degree measure by
Multiply radian measure by
π radians
180°
180°
π radians
—.
—.
Section 9.2
hsnb_alg2_pe_0902.indd 471
Angles and Radian Measure
471
2/5/15 1:48 PM
Convert Between Degrees and Radians
Convert the degree measure to radians or the radian measure to degrees.
π
b. −—
12
a. 120°
READING
The unit “radians” is often
omitted. For instance, the
π
measure −— radians may
12
π
be written simply as −—.
12
SOLUTION
π radians
a. 120° = 120 degrees —
180 degrees
(
π
π
180°
b. −— = −— radians —
12
12
π radians
)
)(
(
2π
=—
3
)
= −15°
Concept Summary
Degree and Radian Measures of Special Angles
The diagram shows equivalent degree and
radian measures for special angles from
0° to 360° (0 radians to 2π radians).
You may find it helpful to memorize the
equivalent degree and radian measures of
special angles in the first quadrant and for
π
90° = — radians. All other special angles
2
shown are multiples of these angles.
Monitoring Progress
5π
6
π
7π
6
y
π
2
radian
π measure
3 π
4
90°
π
120°
60°
6
135°
45°
30°
150° degree
2π
3π 3
4
180°
measure
0°
360°
0 x
2π
210°
330°
225°
315°
11π
240°
300°
6
270°
5π
7π
4 4π
4
5π
3π
3
3
2
Help in English and Spanish at BigIdeasMath.com
Convert the degree measure to radians or the radian measure to degrees.
5π
9. 135°
10. −40°
11. —
12. −6.28
4
A sector is a region of a circle that is bounded by two radii and an arc of the circle.
The central angle θ of a sector is the angle formed by the two radii. There are simple
formulas for the arc length and area of a sector when the central angle is measured
in radians.
Core Concept
Arc Length and Area of a Sector
The arc length s and area A of a sector with
radius r and central angle θ (measured in
radians) are as follows.
sector
r
Arc length: s = rθ
Area: A = —12 r 2θ
472
Chapter 9
hsnb_alg2_pe_0902.indd 472
central
angle θ
arc
length
s
Trigonometric Ratios and Functions
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Modeling with Mathematics
A softball field forms a sector with the dimensions shown. Find the length of the
outfield fence and the area of the field.
SOLUTION
1. Understand the Problem You are given the
dimensions of a softball field. You are asked
to find the length of the outfield fence and the
area of the field.
outfield
fence
200 ft
2. Make a Plan Find the measure of the central
angle in radians. Then use the arc length and
area of a sector formulas.
90°
3. Solve the Problem
200 ft
Step 1 Convert the measure of the central angle to radians.
π radians
90° = 90 degrees —
180 degrees
(
COMMON ERROR
You must write the
measure of an angle
in radians when using
these formulas for the
arc length and area of
a sector.
)
π
= — radians
2
Step 2 Find the arc length and the area of the sector.
1
Area: A = —r 2θ
2
Arc length: s = r θ
π
= 200 —
2
π
1
= — (200)2 —
2
2
= 100π
= 10,000π
≈ 314
≈ 31,416
( )
ANOTHER WAY
Because the central
angle is 90°, the sector
represents —14 of a circle
with a radius of 200 feet.
So,
s = —14
The length of the outfield fence is about 314 feet. The area of the field
is about 31,416 square feet.
4. Look Back To check the area of the field,
consider the square formed using the two
200-foot sides.
⋅ 2πr = — ⋅ 2π (200)
1
4
= 100π
By drawing the diagonal, you can see that
the area of the field is less than the area of the
square but greater than one-half of the area of
the square.
and
A = —41
⋅ πr
2
= —14
⋅ π (200)
= 10,000π.
( )
2
1
—2
⋅ (area of square)
1
2
200 ft
area of square
?
90°
200 ft
?
— (200)2 < 31,416 < 2002
20,000 < 31,416 < 40,000
Monitoring Progress
✓
Help in English and Spanish at BigIdeasMath.com
13. WHAT IF? In Example 4, the outfield fence is 220 feet from home plate. Estimate
the length of the outfield fence and the area of the field.
Section 9.2
hsnb_alg2_pe_0902.indd 473
Angles and Radian Measure
473
2/5/15 1:49 PM
9.2
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE An angle is in standard position when its vertex is at the __________
and its __________ lies on the positive x-axis.
2. WRITING Explain how the sign of an angle measure determines its direction of rotation.
3. VOCABULARY In your own words, define a radian.
4. WHICH ONE DOESN’T BELONG? Which angle does not belong with the other three? Explain
your reasoning.
−90°
450°
−270°
90°
Monitoring Progress and Modeling with Mathematics
In Exercises 5–8, draw an angle with the given measure
in standard position. (See Example 1.)
5. 110°
6. 450°
7. −900°
8. −10°
22. OPEN-ENDED Using radian measure, give one positive
angle and one negative angle that are coterminal with
the angle shown. Justify your answers.
y
In Exercises 9–12, find one positive angle and one
negative angle that are coterminal with the given angle.
(See Example 2.)
9. 70°
x
315°
10. 255°
11. −125°
12. −800°
ANALYZING RELATIONSHIPS In Exercises 23–26, match
the angle measure with the angle.
In Exercises 13–20, convert the degree measure
to radians or the radian measure to degrees.
(See Example 3.)
23. 600°
13. 40°
14. 315°
25. —
15. −260°
16. −500°
π
9
9π
4
24. −—
5π
6
A.
26. −240°
B.
y
y
3π
4
17. —
18. —
19. −5
20. 12
x
x
21. WRITING The terminal side of an angle in
standard position rotates one-sixth of a revolution
counterclockwise from the positive x-axis. Describe
how to find the measure of the angle in both degree
and radian measures.
474
Chapter 9
hsnb_alg2_pe_0902.indd 474
C.
D.
y
x
y
x
Trigonometric Ratios and Functions
2/5/15 1:49 PM
27. MODELING WITH MATHEMATICS The observation
deck of a building forms a sector with the dimensions
shown. Find the length of the safety rail and the area
of the deck. (See Example 4.)
10 yd
10 yd
safety rail
31. PROBLEM SOLVING When a CD player reads
information from the outer edge of a CD, the CD
spins about 200 revolutions per minute. At that speed,
through what angle does a point on the CD spin in
one minute? Give your answer in both degree and
radian measures.
32. PROBLEM SOLVING You work every Saturday from
90°
28. MODELING WITH MATHEMATICS In the men’s shot
put event at the 2012 Summer Olympic Games, the
length of the winning shot was 21.89 meters. A shot
put must land within a sector having a central angle
of 34.92° to be considered fair.
9:00 a.m. to 5:00 p.m. Draw a diagram that shows the
rotation completed by the hour hand of a clock during
this time. Find the measure of the angle generated by
the hour hand in both degrees and radians. Compare
this angle with the angle generated by the minute
hand from 9:00 a.m. to 5:00 p.m.
USING TOOLS In Exercises 33–38, use a calculator to
evaluate the trigonometric function.
4π
3
34. sin —
7π
8
35. csc —
10π
11
36. cot −—
37. cot(−14)
38. cos 6
33. cos —
( 65π )
39. MODELING WITH MATHEMATICS The rear windshield
wiper of a car rotates 120°, as shown. Find the area
cleared by the wiper.
a. The officials draw an arc across the fair landing
area, marking the farthest throw. Find the length
of the arc.
b. All fair throws in the 2012 Olympics landed
within a sector bounded by the arc in part (a).
What is the area of this sector?
25 in.
120°
14 in.
29. ERROR ANALYSIS Describe and correct the error in
converting the degree measure to radians.
✗
(
180 degrees
24° = 24 degrees ——
π radians
=
40. MODELING WITH MATHEMATICS A scientist
)
4320
radians
—
π
performed an experiment to study the effects of
gravitational force on humans. In order for humans
to experience twice Earth’s gravity, they were placed
in a centrifuge 58 feet long and spun at a rate of about
15 revolutions per minute.
≈ 1375.1 radians
30. ERROR ANALYSIS Describe and correct the error
in finding the area of a sector with a radius of
6 centimeters and a central angle of 40°.
✗
1
A = — (6)2(40) = 720 cm2
2
a. Through how many radians did the people rotate
each second?
b. Find the length of the arc through which the
people rotated each second.
Section 9.2
hsnb_alg2_pe_0902.indd 475
Angles and Radian Measure
475
2/5/15 1:49 PM
41. REASONING In astronomy, the terminator is the
day-night line on a planet that divides the planet into
daytime and nighttime regions. The terminator moves
across the surface of a planet as the planet rotates.
It takes about 4 hours for Earth’s terminator to move
across the continental United States. Through what
angle has Earth rotated during this time? Give your
answer in both degree and radian measures.
44. THOUGHT PROVOKING π is an irrational number,
which means that it cannot be written as the ratio
of two whole numbers. π can, however, be written
exactly as a continued fraction, as follows.
1
3 + ————
1
7 + ———
1
15 + ———
1
1 + ——
1
292 + ——
1
1 + ——
1
1+—
1+...
terminator
Show how to use this continued fraction to obtain a
decimal approximation for π.
45. MAKING AN ARGUMENT Your friend claims that
when the arc length of a sector equals the radius, the
s2
area can be given by A = —. Is your friend correct?
2
Explain.
42. HOW DO YOU SEE IT? Use the graph to find the
measure of θ. Explain your reasoning.
y
46. PROBLEM SOLVING A spiral staircase has 15 steps.
4
r=4
Each step is a sector with a radius of 42 inches and a
π
central angle of —.
8
θ
x
a. What is the length of the arc formed by the outer
edge of a step?
b. Through what angle would you rotate by climbing
the stairs?
c. How many square inches of carpeting would you
need to cover the 15 steps?
43. MODELING WITH MATHEMATICS A dartboard is
divided into 20 sectors. Each sector is worth a point
value from 1 to 20 and has shaded regions that double
or triple this value. A sector is shown below. Find the
areas of the entire sector, the double region, and the
triple region.
3 in.
8
3
3 4 in.
triple
47. MULTIPLE REPRESENTATIONS There are 60 minutes
in 1 degree of arc, and 60 seconds in 1 minute of arc.
The notation 50° 30′ 10″ represents an angle with a
measure of 50 degrees, 30 minutes, and 10 seconds.
3 in.
8
2 8 in.
a. Write the angle measure 70.55° using the
notation above.
double
b. Write the angle measure 110° 45′ 30″ to the
nearest hundredth of a degree. Justify your answer.
1
6 5 in.
8
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Find the distance between the two points. (Skills Review Handbook)
48. (1, 4), (3, 6)
49. (−7, −13), (10, 8)
50. (−3, 9), (−3, 16)
51. (2, 12), (8, −5)
52. (−14, −22), (−20, −32)
53. (4, 16), (−1, 34)
476
Chapter 9
hsnb_alg2_pe_0902.indd 476
Trigonometric Ratios and Functions
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Trigonometric Functions of
Any Angle
9.3
Essential Question
How can you use the unit circle to define the
trigonometric functions of any angle?
Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and
—
r = √ x2 + y2 ≠ 0. The six trigonometric functions of θ are defined as shown.
y
sin θ = —
r
r
csc θ = —, y ≠ 0
y
x
cos θ = —
r
r
sec θ = —, x ≠ 0
x
y
tan θ = —, x ≠ 0
x
x
cot θ = —, y ≠ 0
y
y
(x, y)
r
θ
x
Writing Trigonometric Functions
Work with a partner. Find the sine, cosine, and tangent of the angle θ in standard
position whose terminal side intersects the unit circle at the point (x, y) shown.
a.
(
−1 , 3
2 2
(
y
b.
(−12 , 12 (
c.
y
x
y
x
x
(0, −1)
d.
e.
y
f.
y
y
(−1, 0)
x
x
x
( 12 , − 2 3 (
( 12 , −12 (
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in
math, you need to
understand and use stated
assumptions, definitions,
and previously established
results.
Communicate Your Answer
2. How can you use the unit circle to define the trigonometric functions of any angle?
3. For which angles are each function undefined? Explain your reasoning.
a. tangent
b. cotangent
Section 9.3
hsnb_alg2_pe_0903.indd 477
c. secant
d. cosecant
Trigonometric Functions of Any Angle
477
2/5/15 1:49 PM
9.3 Lesson
What You Will Learn
Evaluate trigonometric functions of any angle.
Find and use reference angles to evaluate trigonometric functions.
Core Vocabul
Vocabulary
larry
unit circle, p. 479
quadrantal angle, p. 479
reference angle, p. 480
Previous
circle
radius
Pythagorean Theorem
Trigonometric Functions of Any Angle
You can generalize the right-triangle definitions of trigonometric functions so that they
apply to any angle in standard position.
Core Concept
General Definitions of Trigonometric Functions
Let θ be an angle in standard position, and let (x, y)
be the point where the terminal side of θ intersects
the circle x2 + y2 = r2. The six trigonometric
functions of θ are defined as shown.
y
sin θ = —
r
x
cos θ = —
r
y
tan θ = —, x ≠ 0
x
y
θ
(x, y)
r
r
csc θ = —, y ≠ 0
y
r
sec θ = —, x ≠ 0
x
x
cot θ = —, y ≠ 0
y
x
These functions are sometimes called circular functions.
Evaluating Trigonometric Functions Given a Point
Let (−4, 3) be a point on the terminal side of
an angle θ in standard position. Evaluate the
six trigonometric functions of θ.
y
θ
(−4, 3)
SOLUTION
r
Use the Pythagorean Theorem to find the length of r.
x
—
r = √ x2 + y2
—
= √ (−4)2 + 32
—
= √ 25
=5
Using x = −4, y = 3, and r = 5, the values of the six trigonometric functions of θ are:
478
Chapter 9
hsnb_alg2_pe_0903.indd 478
y 3
sin θ = — = —
r 5
r 5
csc θ = — = —
y 3
4
x
cos θ = — = −—
r
5
5
r
sec θ = — = −—
x
4
3
y
tan θ = — = −—
x
4
4
x
cot θ = — = −—
y
3
Trigonometric Ratios and Functions
2/5/15 1:49 PM
Core Concept
The Unit Circle
y
The circle + = 1, which has center (0, 0)
and radius 1, is called the unit circle. The values
of sin θ and cos θ are simply the y-coordinate and
x-coordinate, respectively, of the point where the
terminal side of θ intersects the unit circle.
x2
y −r
sin θ = — = — = −1.
r
r
The other functions can be
evaluated similarly.
θ
x
r=1
y y
sin θ = — = — = y
r 1
x x
cos θ = — = — = x
r 1
ANOTHER WAY
The general circle
x2 + y2 = r2 can also be used
to find the six trigonometric
functions of θ. The terminal
side of θ intersects the circle
at (0, −r). So,
y2
(x, y)
It is convenient to use the unit circle to find trigonometric functions of quadrantal
angles. A quadrantal angle is an angle in standard position whose terminal side lies on
π
an axis. The measure of a quadrantal angle is always a multiple of 90º, or — radians.
2
Using the Unit Circle
Use the unit circle to evaluate the six trigonometric functions of θ = 270º.
SOLUTION
y
Step 1 Draw a unit circle with the angle θ = 270º in
standard position.
θ
Step 2 Identify the point where the terminal side
of θ intersects the unit circle. The terminal
side of θ intersects the unit circle at (0, −1).
x
Step 3 Find the values of the six trigonometric
functions. Let x = 0 and y = −1 to evaluate
the trigonometric functions.
(0, −1)
y −1
sin θ = — = — = −1
r
1
1
r
csc θ = — = — = −1
y −1
x 0
cos θ = — = — = 0
r 1
r 1
sec θ = — = —
x 0
y −1
tan θ = — = —
x
0
0
x
cot θ = — = — = 0
y −1
undefined
Monitoring Progress
undefined
Help in English and Spanish at BigIdeasMath.com
Evaluate the six trigonometric functions of θ.
1.
2.
y
(−8, 15)
3.
y
θ
θ
x
(3, −3)
y
θ
x
x
(−5, −12)
4. Use the unit circle to evaluate the six trigonometric functions of θ = 180º.
Section 9.3
hsnb_alg2_pe_0903.indd 479
Trigonometric Functions of Any Angle
479
2/5/15 1:49 PM
Reference Angles
Core Concept
READING
Reference Angle Relationships
The symbol θ′ is read as
“theta prime.”
Let θ be an angle in standard position. The reference angle for θ is the acute
angle θ′ formed by the terminal side of θ and the x-axis. The relationship between
θ and θ′ is shown below for nonquadrantal angles θ such that 90° < θ < 360° or,
π
in radians, — < θ < 2π.
2
y
θ′
Quadrant IV
Quadrant III
Quadrant II
y
y
θ
θ
θ
x
Degrees: θ ′ = 180° − θ
Radians: θ ′ = π − θ
θ′
x
Degrees: θ ′ = θ − 180°
Radians: θ ′ = θ − π
θ′
x
Degrees: θ ′ = 360° − θ
Radians: θ ′ = 2π − θ
Finding Reference Angles
5π
Find the reference angle θ ′ for (a) θ = — and (b) θ = −130º.
3
SOLUTION
a. The terminal side of θ lies in Quadrant IV. So,
y
x
θ′
θ
y
5π π
θ′ = 2π − — = —. The figure at the right shows
3
3
π
5π
θ = — and θ′ = —.
3
3
b. Note that θ is coterminal with 230º, whose terminal side
lies in Quadrant III. So, θ′ = 230º − 180º = 50º. The
figure at the left shows θ = −130º and θ′ = 50º.
θ
x
θ′
Reference angles allow you to evaluate a trigonometric function for any angle θ. The
sign of the trigonometric function value depends on the quadrant in which θ lies.
Core Concept
Evaluating Trigonometric Functions
Use these steps to evaluate a
trigonometric function for any angle θ:
Step 1 Find the reference angle θ′.
Step 2 Evaluate the trigonometric
function for θ′.
Step 3 Determine the sign of the
trigonometric function value
from the quadrant in which
θ lies.
480
Chapter 9
hsnb_alg2_pe_0903.indd 480
Signs of Function Values
Quadrant II
sin θ, csc θ : +
cos θ , sec θ : −
tan θ , cot θ : −
Quadrant III
sin θ, csc θ : −
cos θ , sec θ : −
tan θ , cot θ : +
Quadrant I
sin θ, csc θ : +
cos θ , sec θ : +
tan θ , cot θ : +
y
Quadrant IV x
sin θ, csc θ : −
cos θ , sec θ : +
tan θ , cot θ : −
Trigonometric Ratios and Functions
2/5/15 1:49 PM
Using Reference Angles to Evaluate Functions
17π
Evaluate (a) tan(−240º) and (b) csc —.
6
SOLUTION
y
a. The angle −240º is coterminal with 120º. The reference
angle is θ′ = 180º − 120º = 60º. The tangent function θ′ = 60°
is negative in Quadrant II, so
—
x
tan(−240º) = −tan 60º = −√ 3 .
θ = −240°
5π
17π
b. The angle — is coterminal with —. The
6
6
reference angle is
y
5π π
θ′ = π − — = —.
6
6
The cosecant function is positive in Quadrant II, so
INTERPRETING
MODELS
This model neglects air
resistance and assumes
that the projectile’s
starting and ending
heights are the same.
θ′= π6
17π
θ=
6
17π
π
csc — = csc — = 2.
6
6
x
Solving a Real-Life Problem
The horizontal distance d (in feet) traveled by a projectile launched at
an angle θ and with an initial speed v (in feet per second) is given by
v2
d = — sin 2θ.
Model for horizontal distance
32
Estimate the horizontal distance traveled by a golf ball
that is hit at an angle of 50° with an initial speed of
105 feet per second.
50°
SOLUTION
Note that the golf ball is launched at an angle of θ = 50º with initial speed
of v = 105 feet per second.
v2
d = — sin 2θ
32
1052
= — sin(2 50°)
32
Write model for horizontal distance.
⋅
Substitute 105 for v and 50º for θ.
≈ 339
Use a calculator.
The golf ball travels a horizontal distance of about 339 feet.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Sketch the angle. Then find its reference angle.
−7π
9
Evaluate the function without using a calculator.
6. −260°
5. 210°
7. —
15π
4
8. —
11π
4
11. Use the model given in Example 5 to estimate the horizontal distance traveled
by a track and field long jumper who jumps at an angle of 20° and with an initial
speed of 27 feet per second.
9. cos(−210º)
Section 9.3
hsnb_alg2_pe_0903.indd 481
10. sec —
Trigonometric Functions of Any Angle
481
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Exercises
9.3
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE A(n) ___________ is an angle in standard position whose terminal
side lies on an axis.
2. WRITING Given an angle θ in standard position with its terminal side in Quadrant III, explain
how you can use a reference angle to find cos θ.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 –8, evaluate the six trigonometric
functions of θ. (See Example 1.)
3.
4.
y
In Exercises 15–22, sketch the angle. Then find its
reference angle. (See Example 3.)
y
θ
θ
x
x
(5, −12)
(4, −3)
15. −100°
16. 150°
17. 320°
18. −370°
15π
4
19. —
5π
6
21. −—
5.
6.
y
y
θ
(3, 1)
terminal side of an angle θ in standard position.
Describe and correct the error in finding tan θ.
(−6, −8)
7.
8.
y
θ
✗
y
θ
x
13π
6
22. −—
23. ERROR ANALYSIS Let (−3, 2) be a point on the
θ
x
x
8π
3
20. —
x
3
tan θ = — = −—
y
2
24. ERROR ANALYSIS Describe and correct the error in
finding a reference angle θ′ for θ = 650°.
x
(1, −2)
✗
(−12, −9)
θ is coterminal with 290°, whose
terminal side lies in Quadrant IV.
So, θ′ = 290° − 270° = 20°.
In Exercises 9–14, use the unit circle to evaluate the six
trigonometric functions of θ. (See Example 2.)
9. θ = 0°
π
2
10. θ = 540°
7π
2
In Exercises 25–32, evaluate the function without using
a calculator. (See Example 4.)
11. θ = —
12. θ = —
25. sec 135°
26. tan 240°
13. θ = −270°
14. θ = −2π
27. sin(−150°)
28. csc(−420°)
( 34π )
29. tan −—
7π
4
31. cos —
482
Chapter 9
hsnb_alg2_pe_0903.indd 482
( −83 π )
30. cot —
11π
6
32. sec —
Trigonometric Ratios and Functions
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38. REASONING A Ferris wheel has a radius of 75 feet.
In Exercises 33–36, use the model for horizontal
distance given in Example 5.
You board a car at the bottom of the Ferris wheel,
which is 10 feet above the ground, and rotate 255°
counterclockwise before the ride temporarily stops.
How high above the ground are you when the ride
stops? If the radius of the Ferris wheel is doubled,
is your height above the ground doubled? Explain
your reasoning.
33. You kick a football at an angle of 60° with an initial
speed of 49 feet per second. Estimate the horizontal
distance traveled by the football. (See Example 5.)
34. The “frogbot” is a robot designed for exploring rough
terrain on other planets. It can jump at a 45° angle
with an initial speed of 14 feet per second. Estimate
the horizontal distance the frogbot can jump on Earth.
39. DRAWING CONCLUSIONS A sprinkler at ground
level is used to water a garden. The water leaving the
sprinkler has an initial speed of 25 feet per second.
a. Use the model for horizontal distance given in
Example 5 to complete the table.
Angle of
sprinkler, θ
Horizontal distance
water travels, d
30°
35. At what speed must the in-line skater launch himself
35°
off the ramp in order to land on the other side of
the ramp?
40°
45°
50°
55°
60°
18°
5 ft
b. Which value of θ appears to maximize the
horizontal distance traveled by the water? Use the
model for horizontal distance and the unit circle to
explain why your answer makes sense.
36. To win a javelin throwing competition, your last
throw must travel a horizontal distance of at least
100 feet. You release the javelin at a 40° angle with
an initial speed of 71 feet per second. Do you win the
competition? Justify your answer.
37. MODELING WITH MATHEMATICS A rock climber is
using a rock climbing treadmill that is 10 feet long.
The climber begins by lying horizontally on the
treadmill, which is then rotated about its midpoint by
110° so that the rock climber is climbing toward the
top. If the midpoint of the treadmill is 6 feet above
the ground, how high above the ground is the top of
the treadmill?
c. Compare the horizontal distance traveled by the
water when θ = (45 − k)° with the distance when
θ = (45 + k)°, for 0 < k < 45.
40. MODELING WITH MATHEMATICS Your school’s
y
marching band is performing at halftime during
a football game. In the last formation, the band
members form a circle 100 feet wide in the center
of the field. You start at a point on the circle 100 feet
from the goal line, march 300° around the circle, and
then walk toward the goal line to exit the field. How
far from the goal line are you at the point where you
leave the circle?
y
5 ft
110°
x
300°
?
starting
position
(50, 0)
100 ft
6 ft
(x, y)
x
?
goal line
Section 9.3
hsnb_alg2_pe_0903.indd 483
Trigonometric Functions of Any Angle
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41. ANALYZING RELATIONSHIPS Use symmetry and
46. MAKING AN ARGUMENT Your friend claims that
the given information to label the coordinates of the
other points corresponding to special angles on the
unit circle.
(0, 1)
y
( 12 , 23 (
90°
60°
45°
30°
120°
135°
150°
0°
360°
330°
315°
300°
270°
180°
210°
225°
240°
the only solution
to the trigonometric equation
—
tan θ = √3 is θ = 60°. Is your friend correct?
Explain your reasoning.
47. PROBLEM SOLVING When two atoms in a molecule
( 22 , 22 (
( 23 , 12 (
are bonded to a common atom, chemists are interested
in both the bond angle and the lengths of the bonds.
An ozone molecule is made up of two oxygen atoms
bonded to a third oxygen atom, as shown.
x
(1, 0)
y
(x, y)
d
128 pm
117°
(0, 0)
42. THOUGHT PROVOKING Use the interactive unit circle
tool at BigIdeasMath.com to describe all values of θ
for each situation.
x
128 pm
(128, 0)
a. In the diagram, coordinates are given in
picometers (pm). (Note: 1 pm = 10−12 m) Find the
coordinates (x, y) of the center of the oxygen atom
in Quadrant II.
a. sin θ > 0, cos θ < 0, and tan θ > 0
b. sin θ > 0, cos θ < 0, and tan θ < 0
b. Find the distance d (in picometers) between the
centers of the two unbonded oxygen atoms.
43. CRITICAL THINKING Write tan θ as the ratio of two
other trigonometric functions. Use this ratio to explain
why tan 90° is undefined but cot 90° = 0.
48. MATHEMATICAL CONNECTIONS The latitude of a
point on Earth is the degree measure of the shortest
arc from that point to the equator. For example,
the latitude of point P in the diagram equals the
degree measure of arc PE. At what latitude θ is the
circumference of the circle of latitude at P half the
distance around the equator?
44. HOW DO YOU SEE IT? Determine whether each
of the six trigonometric functions of θ is positive,
negative, or zero. Explain your reasoning.
y
θ
circle of
latitude
x
O
45. USING STRUCTURE A line with slope m passes
through the origin. An angle θ in standard position
has a terminal side that coincides with the line. Use
a trigonometric function to relate the slope of the line
to the angle.
Maintaining Mathematical Proficiency
P
C
θ
D
E
equator
Reviewing what you learned in previous grades and lessons
Find all real zeros of the polynomial function. (Section 4.6)
49. f (x) = x4 + 2x3 + x2 + 8x − 12
50. f (x) = x5 + 4x4 − 14x3 − 14x2 − 15x − 18
Graph the function. (Section 4.8)
51. f (x) = 2(x + 3)2(x − 1)
484
Chapter 9
hsnb_alg2_pe_0903.indd 484
1
52. f (x) = —3 (x − 4)(x + 5)(x + 9)
53. f (x) = x2(x + 1)3(x − 2)
Trigonometric Ratios and Functions
2/5/15 1:49 PM
9.4
Graphing Sine and Cosine Functions
Essential Question
What are the characteristics of the graphs of the
sine and cosine functions?
Graphing the Sine Function
Work with a partner.
a. Complete the table for y = sin x, where x is an angle measure in radians.
x
−2π
7π
−—
4
π
4
—
3π − 5π −π − 3π −π
−—
—
—
—
2
4
4
2
π
−—
4
0
2π
—
y = sin x
—
x
π
2
3π
4
—
π
5π
4
—
3π
2
—
7π
4
—
9π
4
y = sin x
b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to
sketch the graph of y = sin x.
y
1
−2π
π
−3
2
−π
π
−
2
π
2
π
3π
2
2π
5π x
2
−1
c. Use the graph to identify the x-intercepts, the x-values where the local maximums
and minimums occur, and the intervals for which the function is increasing or
decreasing over −2π ≤ x ≤ 2π. Is the sine function even, odd, or neither?
Graphing the Cosine Function
Work with a partner.
a. Complete a table for y = cos x using the same values of x as those used in
Exploration 1.
b. Plot the points (x, y) from part (a) and sketch the graph of y = cos x.
LOOKING FOR
STRUCTURE
To be proficient in math,
you need to look closely
to discern a pattern
or structure.
c. Use the graph to identify the x-intercepts, the x-values where the local maximums
and minimums occur, and the intervals for which the function is increasing or
decreasing over −2π ≤ x ≤ 2π. Is the cosine function even, odd, or neither?
Communicate Your Answer
3. What are the characteristics of the graphs of the sine and cosine functions?
4. Describe the end behavior of the graph of y = sin x.
Section 9.4
hsnb_alg2_pe_0904.indd 485
Graphing Sine and Cosine Functions
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9.4 Lesson
What You Will Learn
Explore characteristics of sine and cosine functions.
Stretch and shrink graphs of sine and cosine functions.
Core Vocabul
Vocabulary
larry
amplitude, p. 486
periodic function, p. 486
cycle, p. 486
period, p. 486
phase shift, p. 488
midline, p. 488
Previous
transformations
x-intercept
Translate graphs of sine and cosine functions.
Reflect graphs of sine and cosine functions.
Exploring Characteristics of Sine and Cosine Functions
In this lesson, you will learn to graph sine and cosine functions. The graphs of sine
and cosine functions are related to the graphs of the parent functions y = sin x and
y = cos x, which are shown below.
x
3π
−2π − —
2
−π
π
−—
2
0
—
π
2
π
—
3π
2
2π
y = sin x
0
1
0
−1
0
1
0
−1
0
y = cos x
1
0
−1
0
1
0
−1
0
1
y
maximum
value: 1
y = sin x
1
amplitude: 1
range:
−1 ≤ y ≤ 1
−
3π −π
2
π
−
2
π
2
−1
range:
−1 ≤ y ≤ 1
3π
2
2π
x
period:
2π
minimum
value: −1
maximum
m
value: 1 y = cos x
π
y
amplitude: 1
− 2π
−
3π −π
2
π
−
2
minimum
value: −1
−1
π
2
π
3π
2
2π
x
period:
2π
Core Concept
Characteristics of y = sin x and y = cos x
• The domain of each function is all real numbers.
• The range of each function is −1 ≤ y ≤ 1. So, the minimum value of each
function is −1 and the maximum value is 1.
• The amplitude of the graph of each function is one-half of the difference of
the maximum value and the minimum value, or —12 [1 − (−1)] = 1.
• Each function is periodic, which means that its graph has a repeating pattern.
The shortest repeating portion of the graph is called a cycle. The horizontal
length of each cycle is called the period. Each graph shown above has a period
of 2π.
• The x-intercepts for y = sin x occur when x = 0, ±π, ±2π, ±3π, . . ..
π 3π 5π 7π
• The x-intercepts for y = cos x occur when x = ± —, ± —, ± —, ± —, . . ..
2
2
2
2
486
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hsnb_alg2_pe_0904.indd 486
Trigonometric Ratios and Functions
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Stretching and Shrinking Sine and Cosine Functions
REMEMBER
⋅
The graph of y = a f (x) is
a vertical stretch or shrink
of the graph of y = f (x) by
a factor of a.
The graphs of y = a sin bx and y = a cos bx represent transformations of their parent
functions. The value of a indicates a vertical stretch (a > 1) or a vertical shrink
(0 < a < 1) and changes the amplitude of the graph. The value of b indicates a
horizontal stretch (0 < b < 1) or a horizontal shrink (b > 1) and changes the period
of the graph.
y = a sin bx
y = a cos bx
1
horizontal stretch or shrink by a factor of —
b
vertical stretch or shrink by a factor of a
The graph of y = f (bx)
is a horizontal stretch or
shrink of the graph of
1
y = f (x) by a factor of —.
b
Core Concept
Amplitude and Period
The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where
a and b are nonzero real numbers, are as follows:
2π
Period = —
∣b∣
Amplitude = ∣ a ∣
2π
Each graph below shows five key points that partition the interval 0 ≤ x ≤ — into
b
four equal parts. You can use these points to sketch the graphs of y = a sin bx and
y = a cos bx. The x-intercepts, maximum, and minimum occur at these points.
y
( 14 ∙ 2bπ , a(
y
y = a sin bx
(0, a)
( 2bπ , 0(
(0, 0)
y = a cos bx
x
( 12 ∙ 2bπ , 0(
( 2bπ , a(
( 14 ∙ 2bπ , 0(
( 34 ∙ 2bπ , 0(
x
( 12 ∙ 2bπ , −a(
( 34 ∙ 2bπ , −a(
Graphing a Sine Function
Identify the amplitude and period of g(x) = 4 sin x. Then graph the function and
describe the graph of g as a transformation of the graph of f (x) = sin x.
REMEMBER
A vertical stretch of a
graph does not change its
x-intercept(s). So, it makes
sense that the x-intercepts
of g(x) = 4 sin x and
f (x) = sin x are the same.
4
g
−
f
π
4
9π
4
SOLUTION
The function is of the form g(x) = a sin bx where a = 4 and b = 1. So, the amplitude
2π 2π
is a = 4 and the period is — = — = 2π.
b
1
(
)
π
1
Maximum: — 2π, 4 = —, 4
4
2
(⋅ ) ( )
3
3π
Minimum: ( ⋅ 2π, −4 ) = ( , −4 )
4
2
—
−4
⋅
1
Intercepts: (0, 0); — 2π, 0 = (π, 0); (2π, 0)
2
y
4
π
2
3π
2
x
—
The graph of g is a vertical stretch by a factor of 4 of the graph of f.
Section 9.4
hsnb_alg2_pe_0904.indd 487
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Graphing a Cosine Function
1
Identify the amplitude and period of g(x) = — cos 2πx. Then graph the function and
2
describe the graph of g as a transformation of the graph of f (x) = cos x.
SOLUTION
1
The function is of the form g(x) = a cos bx where a = — and b = 2π. So, the
2
2π 2π
1
amplitude is a = — and the period is — = — = 1.
2
b
2π
STUDY TIP
After you have drawn
one complete cycle of
the graph in Example 2
on the interval 0 ≤ x ≤ 1,
you can extend the graph
by repeating the cycle as
many times as desired to
the left and right of
0 ≤ x ≤ 1.
(
) ( )( ⋅ ) ( )
1
1
Maximums: ( 0, ); ( 1, )
2
2
1
1 1
1
Minimum: ( ⋅ 1, − ) = ( , − )
2
2
2 2
⋅
1
1
3
3
Intercepts: — 1, 0 = — , 0 ; — 1, 0 = — , 0
4
4
4
4
—
—
y
—
—
1
—
—
1
2 x
−1
1
The graph of g is a vertical shrink by a factor of — and a horizontal shrink by a
2
1
factor of — of the graph of f.
2π
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Identify the amplitude and period of the function. Then graph the function and
describe the graph of g as a transformation of the graph of its parent function.
REMEMBER
1
The graph of y = f (x) + k
is a vertical translation of
the graph of y = f (x).
The graph of y = f (x − h)
is a horizontal translation
of the graph of y = f (x).
1. g(x) = —4 sin x
2. g(x) = cos 2x
3. g(x) = 2 sin πx
1
1
4. g(x) = —3 cos —2 x
Translating Sine and Cosine Functions
The graphs of y = a sin b(x − h) + k and y = a cos b(x − h) + k represent
translations of y = a sin bx and y = a cos bx. The value of k indicates a translation up
(k > 0) or down (k < 0). The value of h indicates a translation left (h < 0) or right
(h > 0). A horizontal translation of a periodic function is called a phase shift.
Core Concept
Graphing y = a sin b(x − h) + k and y = a cos b(x − h) + k
To graph y = a sin b(x − h) + k or y = a cos b(x − h) + k where a > 0 and
b > 0, follow these steps:
2π
Step 1 Identify the amplitude a, the period —, the horizontal shift h, and the
b
vertical shift k of the graph.
Step 2 Draw the horizontal line y = k, called the midline of the graph.
Step 3 Find the five key points by translating the key points of y = a sin bx or
y = a cos bx horizontally h units and vertically k units.
Step 4 Draw the graph through the five translated key points.
488
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hsnb_alg2_pe_0904.indd 488
Trigonometric Ratios and Functions
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Graphing a Vertical Translation
Graph g(x) = 2 sin 4x + 3.
LOOKING FOR
STRUCTURE
The graph of g is a
translation 3 units up of
the graph of f (x) = 2 sin 4x.
So, add 3 to the
y-coordinates of the
five key points of f.
SOLUTION
Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.
Amplitude: a = 2
Horizontal shift: h = 0
2π 2π π
Period: — = — = —
b
4
2
Vertical shift: k = 3
Step 2 Draw the midline of the graph, y = 3.
Step 3 Find the five key points.
π
π
π
π
On y = k: (0, 0 + 3) = (0, 3); —, 0 + 3 = —, 3 ; —, 0 + 3 = —, 3
4
4
2
2
(
π
π
Maximum: ( , 2 + 3 ) = ( , 5 )
8
8
—
) ( )(
—
y
5
3π
3π
Minimum: —, −2 + 3 = —, 1
8
8
(
) ( )
) ( )
1
Step 4 Draw the graph through the key points.
π
4
−1
π
2
x
Graphing a Horizontal Translation
1
Graph g(x) = 5 cos — (x − 3π).
2
LOOKING FOR
STRUCTURE
The graph of g is a
translation 3π units
right of the graph of
f (x) = 5 cos —12 x. So, add 3π
to the x-coordinates of the
five key points of f.
SOLUTION
Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.
Horizontal shift: h = 3π
Amplitude: a = 5
2π 2π
Period: — = — = 4π
b
1
—
2
Vertical shift: k = 0
Step 2 Draw the midline of the graph. Because k = 0, the midline is the x-axis.
Step 3 Find the five key points.
y
On y = k: (π + 3π, 0) = (4π, 0);
(3π + 3π, 0) = (6π, 0)
6
2
Maximums: (0 + 3π, 5) = (3π, 5);
(4π + 3π, 5) = (7π, 5)
−2
Minimum: (2π + 3π, −5) = (5π, −5)
3π
5π
7π
9π
−6
Step 4 Draw the graph through the key points.
Monitoring Progress
x
π
Help in English and Spanish at BigIdeasMath.com
Graph the function.
5. g(x) = cos x + 4
Section 9.4
hsnb_alg2_pe_0904.indd 489
1
2
(
π
2
6. g(x) = — sin x − —
)
7. g(x) = sin(x + π) − 1
Graphing Sine and Cosine Functions
489
2/5/15 1:50 PM
Reflecting Sine and Cosine Functions
You have graphed functions of the form y = a sin b(x − h) + k and
y = a cos b(x − h) + k, where a > 0 and b > 0. To see what happens when a < 0,
consider the graphs of y = −sin x and y = −cos x.
y
y = −sin x
1
REMEMBER
This result makes sense
because the graph of
y = −f (x) is a reflection in
the x-axis of the graph of
y = f (x).
π
2
−1
1
(2π, 0)
(0, 0)
y = −cos x
y
( 32π , 1(
(π, 1)
( 32π , 0(
( π2, 0(
π
x
(π, 0)
(0, −1)
( π2 , −1(
x
2π
(2π, −1)
The graphs are reflections of the graphs of y = sin x and y = cos x in the x-axis. In
general, when a < 0, the graphs of y = a sin b(x − h) + k and y = a cos b(x − h) + k
are reflections of the graphs of y = ∣ a ∣ sin b(x − h) + k and y = ∣ a ∣ cos b(x − h) + k,
respectively, in the midline y = k.
Graphing a Reflection
π
2
Graph g(x) = −2 sin — x − — .
3
2
)
(
SOLUTION
Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.
π
Horizontal shift: h = —
2
Amplitude: ∣ a ∣ = ∣ −2 ∣ = 2
2π 2π
Period: — = — = 3π
b
2
—
3
Vertical shift: k = 0
Step 2 Draw the midline of the graph. Because k = 0, the midline is the x-axis.
π
2
Step 3 Find the five key points of f (x) = ∣ −2 ∣ sin — x − — .
3
2
π
π
π
3π π
7π
On y = k: 0 + —, 0 = —, 0 ; — + —, 0 = (2π, 0); 3π + —, 0 = —, 0
2
2
2
2
2
2
(
) ( )(
In Example 5, the
maximum value and
minimum value of f
are the minimum value
and maximum value,
respectively, of g.
)
)
3π π
5π
9π π
11π
, −2 )
Maximum: (
+ , 2) = ( , 2)
Minimum: (
+ , −2 ) = (
4
2
4
4
2
4
(
STUDY TIP
)
—
—
—
—
Step 4 Reflect the graph. Because a < 0,
the graph is reflected in the midline
5π
5π
y = 0. So, —, 2 becomes —, −2
4
4
11π
11π
and —, −2 becomes —, 2 .
4
4
( )
(
)
(
(
) (
(
—
—
y
)
)
1
−1
π
3π
x
Step 5 Draw the graph through the key points.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Graph the function.
(
π
2
8. g(x) = −cos x + —
490
Chapter 9
hsnb_alg2_pe_0904.indd 490
)
1
2
9. g(x) = −3 sin — x + 2
10. g(x) = −2 cos 4x − 1
Trigonometric Ratios and Functions
2/5/15 1:50 PM
9.4
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE The shortest repeating portion of the graph of a periodic function is
called a(n) _________.
1
2. WRITING Compare the amplitudes and periods of the functions y = —2 cos x and y = 3 cos 2x.
3. VOCABULARY What is a phase shift? Give an example of a sine function that has a phase shift.
4. VOCABULARY What is the midline of the graph of the function y = 2 sin 3(x + 1) − 2?
Monitoring Progress and Modeling with Mathematics
USING STRUCTURE In Exercises 5–8, determine whether
the graph represents a periodic function. If so, identify
the period.
y
5.
y
6.
1
1
x
2
4
π
2
x
In Exercises 13–20, identify the amplitude and period of
the function. Then graph the function and describe the
graph of g as a transformation of the graph of its parent
function. (See Examples 1 and 2.)
13. g(x) = 3 sin x
14. g(x) = 2 sin x
15. g(x) = cos 3x
16. g(x) = cos 4x
17. g(x) = sin 2π x
18. g(x) = 3 sin 2x
1
19. g(x) = —3 cos 4x
7.
8.
y
y
21. ANALYZING EQUATIONS Which functions have an
4
1
1
20. g(x) = —2 cos 4πx
amplitude of 4 and a period of 2?
2
10 x
A y = 4 cos 2x
○
B y = −4 sin πx
○
−1
2
4
6 x
C y = 2 sin 4x
○
In Exercises 9–12, identify the amplitude and period of
the graph of the function.
y
9.
10.
D y = 4 cos πx
○
22. WRITING EQUATIONS Write an equation of the form
y = a sin bx, where a > 0 and b > 0, so that the graph
has the given amplitude and period.
y
0.5
1
2π
11.
x
1
12.
y
2 x
y
2
π
2
π x
π
4π
7π
x
−4
Section 9.4
hsnb_alg2_pe_0904.indd 491
b. amplitude: 10
period: 4
c. amplitude: 2
period: 2π
d. amplitude: —12
period: 3π
23. MODELING WITH MATHEMATICS The motion
4
−2
a. amplitude: 1
period: 5
of a pendulum can be modeled by the function
d = 4 cos 8π t, where d is the horizontal displacement
(in inches) of the pendulum relative to its position at
rest and t is the time (in seconds). Find and interpret
the period and amplitude in the context of this
situation. Then graph the function.
Graphing Sine and Cosine Functions
491
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24. MODELING WITH MATHEMATICS A buoy bobs up
and down as waves go past. The vertical displacement
y (in feet) of the buoy with respect to sea level can be
π
modeled by y = 1.75 cos —t, where t is the time
3
(in seconds). Find and interpret the period and
amplitude in the context of the problem. Then graph
the function.
USING STRUCTURE In Exercises 37– 40, describe the
transformation of the graph of f represented by the
function g.
( π2 )
π
f (x) = sin x, g(x) = 3 sin( x + ) − 2
4
37. f (x) = cos x, g(x) = 2 cos x − — + 1
38.
—
39. f (x) = sin x, g(x) = sin 3(x + 3π) − 5
40. f (x) = cos x, g(x) = cos 6(x − π) + 9
In Exercises 41–48, graph the function. (See Example 5.)
41. g(x) = −cos x + 3
42. g(x) = −sin x − 5
1
2
43. g(x) = −sin —x − 2
In Exercises 25–34, graph the function. (See Examples
3 and 4.)
25. g(x) = sin x + 2
π
2
(
27. g(x) = cos x − —
)
(
π
4
29. g(x) = 2 cos x − 1
45. g(x) = −sin(x − π) + 4
46. g(x) = −cos(x + π) − 2
26. g(x) = cos x − 4
28. g(x) = sin x + —
)
30. g(x) = 3 sin x + 1
31. g(x) = sin 2(x + π)
( π4 )
π
g(x) = −5 sin( x − ) + 3
2
47. g(x) = −4 cos x + — − 1
48.
—
49. USING EQUATIONS Which of the following is a
point where the maximum value of the graph of
π
y = −4 cos x − — occurs?
2
π
π
−—, 4
A
B —, 4
○
○
2
2
32. g(x) = cos 2(x − π)
(
1
33. g(x) = sin —(x + 2π) + 3
2
(
1
2
)
)
( )
D (π, 4)
○
C (0, 4)
○
34. g(x) = cos —(x − 3π) − 5
35. ERROR ANALYSIS Describe and correct the error in
50. ANALYZING RELATIONSHIPS Match each function
with its graph. Explain your reasoning.
2
finding the period of the function y = sin —x.
3
✗
44. g(x) = −cos 2x + 1
2
∣= 1
∣b∣ ∣ —
3
Period:
=
—
2π —
2π —
3π
a. y = 3 + sin x
b. y = −3 + cos x
π
c. y = sin 2 x − —
2
(
A.
π
d. y = cos 2 x − —
2
)
(
B.
y
π
2
36. ERROR ANALYSIS Describe and correct the error in
(
✗
)
−1
hsnb_alg2_pe_0904.indd 492
x
y
C.
Maximum:
( ( —14 ⋅2π ) − —π2, 2 ) = ( —π2 − —π2, 2 )
Chapter 9
π
1
π
= (0, 2)
492
y
4
1
determining the point where the maximum value of
π
the function y = 2 sin x − — occurs.
2
−1
)
D.
π
2π x
2π x
y
1
x
π
2
π
−4
Trigonometric Ratios and Functions
2/5/15 1:50 PM
WRITING EQUATIONS In Exercises 51–54, write a rule
for g that represents the indicated transformations of
the graph of f.
57. USING TOOLS The average wind speed s (in miles per
hour) in the Boston Harbor can be approximated by
π
s = 3.38 sin — (t + 3) + 11.6
180
51. f (x) = 3 sin x; translation 2 units up and π units right
where t is the time in days and t = 0 represents
January 1. Use a graphing calculator to graph the
function. On which days of the year is the average
wind speed 10 miles per hour? Explain your
reasoning.
52. f (x) = cos 2πx; translation 4 units down and 3 units left
1
53. f (x) = —3 cos πx; translation 1 unit down, followed by
a reflection in the line y = −1
1
58. USING TOOLS The water depth d (in feet) for the Bay
3
54. f (x) = —2 sin 6x; translation —2 units down and 1 unit
π
of Fundy can be modeled by d = 35 − 28 cos —t,
6.2
where t is the time in hours and t = 0 represents
midnight. Use a graphing calculator to graph the
function. At what time(s) is the water depth 7 feet?
Explain.
3
right, followed by a reflection in the line y = −—2
55. MODELING WITH MATHEMATICS The height h
(in feet) of a swing above the ground can be modeled
by the function h = −8 cos θ + 10, where the pivot is
10 feet above the ground, the rope is 8 feet long, and
θ is the angle that the rope makes with the vertical.
Graph the function. What is the height of the swing
when θ is 45°?
8 ft
10 − h 8 ft θ
high tide
low tide
10 ft
h
59. MULTIPLE REPRESENTATIONS Find the average rate of
Front view
change of each function over the interval 0 < x < π.
Side view
a. y = 2 cos x
56. DRAWING A CONCLUSION In a particular region, the
b.
population L (in thousands) of lynx (the predator) and
the population H (in thousands) of hares (the prey)
can be modeled by the equations
x
f (x) = −cos x
π
L = 11.5 + 6.5 sin — t
5
π
H = 27.5 + 17.5 cos — t
5
c.
1
b. Use the figure to explain how the changes in the
two populations appear to be related.
Population
(thousands)
20
2π
−1
0
1
0
−1
f
x
b. Graph each function.
L
16
c. Describe the transformations of the graphs of the
parent functions.
t
Time (years)
Section 9.4
hsnb_alg2_pe_0904.indd 493
3π
2
a. Construct a table of values for each equation
using the quadrantal angles in the interval
−2π ≤ x ≤ 2π.
H
40
12
—
y = cos(−x).
y
8
π
60. REASONING Consider the functions y = sin(−x) and
Animal Populations
4
π
2
π
a. Determine the ratio of hares to lynx when
t = 0, 2.5, 5, and 7.5 years.
0
—
y
where t is the time in years.
0
0
Graphing Sine and Cosine Functions
493
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61. MODELING WITH MATHEMATICS You are riding a
66. THOUGHT PROVOKING Use a graphing calculator
to find a function of the form y = sin b1x + cos b2x
whose graph matches that shown below.
Ferris wheel that turns for 180 seconds. Your height
h (in feet) above the ground at any time t (in seconds)
can be modeled by the equation
π
h = 85 sin —(t − 10) + 90.
20
a. Graph the function.
y
−6
−4
b. How many cycles
does the Ferris wheel
make in 180 seconds?
−2
2
4
6x
−2
c. What are your maximum
and minimum heights?
67. PROBLEM SOLVING For a person at rest, the blood
pressure P (in millimeters of mercury) at time t (in
seconds) is given by the function
62. HOW DO YOU SEE IT? Use the graph to answer
each question.
8π
P = 100 − 20 cos —t.
3
Graph the function. One cycle is equivalent to one
heartbeat. What is the pulse rate (in heartbeats per
minute) of the person?
y
6
−π
π
x
−6
a. Does the graph represent a function of the form
f(x) = a sin bx or f(x) = a cos bx? Explain.
b. Identify the maximum value, minimum value,
period, and amplitude of the function.
68. PROBLEM SOLVING The motion of a spring can
be modeled by y = A cos kt, where y is the vertical
displacement (in feet) of the spring relative to its
position at rest, A is the initial displacement (in feet),
k is a constant that measures the elasticity of the
spring, and t is the time (in seconds).
63. FINDING A PATTERN Write an expression in terms of
the integer n that represents all the x-intercepts of the
graph of the function y = cos 2x. Justify your answer.
64. MAKING AN ARGUMENT Your friend states that for
a. You have a spring whose motion can be modeled
by the function y = 0.2 cos 6t. Find the initial
displacement and the period of the spring. Then
graph the function.
functions of the form y = a sin bx and y = a cos bx,
the values of a and b affect the x-intercepts of the
graph of the function. Is your friend correct? Explain.
b. When a damping force is applied to the spring,
the motion of the spring can be modeled by the
function y = 0.2e−4.5t cos 4t. Graph this function.
What effect does damping have on the motion?
65. CRITICAL THINKING Describe a transformation of the
graph of f (x) = sin x that results in the graph of
g(x) = cos x.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Simplify the rational expression, if possible. (Section 7.3)
x2 + x − 6
x+3
69. —
x3 − 2x2 − 24x
x − 2x − 24
70. ——
2
x2 − 4x − 5
x + 4x − 5
71. —
2
x2 − 16
x + x − 20
72. —
2
Find the least common multiple of the expressions. (Section 7.4)
73. 2x, 2(x − 5)
494
Chapter 9
hsnb_alg2_pe_0904.indd 494
74. x2 − 4, x + 2
75. x2 + 8x + 12, x + 6
Trigonometric Ratios and Functions
2/5/15 1:50 PM
9.1–9.4
What Did You Learn?
Core Vocabulary
sine, p. 462
cosine, p. 462
tangent, p. 462
cosecant, p. 462
secant, p. 462
cotangent, p. 462
initial side, p. 470
terminal side, p. 470
standard position, p. 470
coterminal, p. 471
radian, p. 471
sector, p. 472
central angle, p. 472
unit circle, p. 479
quadrantal angle, p. 479
reference angle, p. 480
amplitude, p. 486
periodic function, p. 486
cycle, p. 486
period, p. 486
phase shift, p. 488
midline, p. 488
Core Concepts
Section 9.1
Right Triangle Definitions of Trigonometric Functions, p. 462
Trigonometric Values for Special Angles, p. 463
Section 9.2
Angles in Standard Position, p. 470
Converting Between Degrees and Radians, p. 471
Degree and Radian Measures of Special Angles, p. 472
Arc Length and Area of a Sector, p. 472
Section 9.3
General Definitions of Trigonometric Functions, p. 478
The Unit Circle, p. 479
Reference Angle Relationships, p. 480
Evaluating Trigonometric Functions, p. 480
Section 9.4
Characteristics of y = sin x and y = cos x, p. 486
Amplitude and Period, p. 487
Graphing y = a sin b(x − h) + k and y = a cos b(x − h) + k, p. 488
Mathematical Practices
1.
Make a conjecture about the horizontal distances traveled in part (c) of Exercise 39 on page 483.
2.
Explain why the quantities in part (a) of Exercise 56 on page 493 make sense in the context of
the situation.
Study Skills
Form a Final Exam
Study Group
Form a study group several weeks before the final exam.
The intent of this group is to review what you have already
learned while continuing to learn new material.
495
hsnb_alg2_pe_09mc.indd 495
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9.1–9.4
Quiz
2
1. In a right triangle, θ is an acute angle and sin θ = —7 . Evaluate the other five trigonometric
functions of θ. (Section 9.1)
Find the value of x for the right triangle. (Section 9.1)
2.
3.
60°
30°
x
8
4.
12
27
x
49°
x
Draw an angle with the given measure in standard position. Then find one positive angle
and one negative angle that are coterminal with the given angle. (Section 9.2)
5π
6
5. 40°
7. −960°
6. —
Convert the degree measure to radians or the radian measure to degrees. (Section 9.2)
3π
10
9. −60°
8. —
10. 72°
Evaluate the six trigonometric functions of θ. (Section 9.3)
11.
12.
y
13.
y
y
θ = π2
θ
x
2π
θ= 3
x
x
(−2, −6)
14. Identify the amplitude and period of g(x) = 3 sin x. Then graph the function and describe
the graph of g as a transformation of the graph of f (x) = sin x. (Section 9.4)
15. Identify the amplitude and period of g(x) = cos 5πx + 3. Then graph the
function and describe the graph of g as a transformation of the graph
of f(x) = cos x. (Section 9.4)
16. You are flying a kite at an angle of 70°. You have let out a total of
400 feet of string and are holding the reel steady 4 feet above the ground.
(Section 9.1)
a. How high above the ground is the kite?
b. A friend watching the kite estimates that the angle of elevation
to the kite is 85°. How far from your friend are you standing?
400 ft
Not drawn to scale
70° 85°
4 ft
17. The top of the Space Needle in Seattle, Washington, is a revolving, circular
restaurant. The restaurant has a radius of 47.25 feet and makes one complete
revolution in about an hour. You have dinner at a window table from 7:00 p.m.
to 8:55 p.m. Compare the distance you revolve with the distance of a person
seated 5 feet away from the windows. (Section 9.2)
496
Chapter 9
hsnb_alg2_pe_09mc.indd 496
Trigonometric Ratios and Functions
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9.5
Graphing Other Trigonometric
Functions
Essential Question
What are the characteristics of the graph of
the tangent function?
Graphing the Tangent Function
Work with a partner.
a. Complete the table for y = tan x, where x is an angle measure in radians.
x
π
−—
2
π
−—
3
π
−—
4
π
−—
6
0
—
π
6
—
π
4
—
π
3
—
2π
3
—
3π
4
—
5π
6
π
—
7π
6
—
5π
4
—
4π
3
—
3π
2
—
π
2
y = tan x
x
—
5π
3
y = tan x
b. The graph of y = tan x has vertical asymptotes at x-values where tan x is undefined.
Plot the points (x, y) from part (a). Then use the asymptotes to sketch the graph of
y = tan x.
y
6
4
2
−
π
2
π
2
π
3π
2
x
−2
−4
−6
MAKING SENSE
OF PROBLEMS
To be proficient in math,
you need to consider
analogous problems and
try special cases of the
original problem in order
to gain insight into
its solution.
c. For the graph of y = tan x, identify the asymptotes, the x-intercepts, and the
π
3π
intervals for which the function is increasing or decreasing over −— ≤ x ≤ —.
2
2
Is the tangent function even, odd, or neither?
Communicate Your Answer
2. What are the characteristics of the graph of the tangent function?
π
2
Section 9.5
hsnb_alg2_pe_0905.indd 497
3π
2
3. Describe the asymptotes of the graph of y = cot x on the interval −— < x < —.
Graphing Other Trigonometric Functions
497
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9.5 Lesson
What You Will Learn
Explore characteristics of tangent and cotangent functions.
Graph tangent and cotangent functions.
Core Vocabul
Vocabulary
larry
Graph secant and cosecant functions.
Previous
asymptote
period
amplitude
x-intercept
transformations
Exploring Tangent and Cotangent Functions
The graphs of tangent and cotangent functions are related to the graphs of the parent
functions y = tan x and y = cot x, which are graphed below.
x
π
−—
2
π
x approaches −—
2
−1.57
−1.5
π
−—
4
0
—
−1
0
1
y = tan x Undef. −1256 −14.10
π
4
tan x approaches −∞
sin x
Because tan x = —, tan x
cos x
is undefined for x-values at
which cos x = 0, such as
π
x = ± — ≈ ±1.571.
2
The table indicates that the graph
has asymptotes at these values.
The table represents one cycle of the
graph, so the period of the graph is π.
You can use a similar approach
to graph y = cot x. Because
cos x
cot x = —, cot x is undefined for
sin x
x-values at which sin x = 0, which
are multiples of π. The graph has
asymptotes at these values. The
period of the graph is also π.
3π
2
1.5
1.57
π
2
—
14.10 1256 Undef.
tan x approaches ∞
y
y = tan x
−
π
x approaches —
2
−π
2
−
π
2
π
2
π
3π x
2
−2
period: π
y
y = cot x
2
−π
−
π
2
π
2
π
3π
2
x
2π
period: π
Core Concept
Characteristics of y = tan x and y = cot x
STUDY TIP
The functions y = tan x and y = cot x have the following characteristics.
π
• The domain of y = tan x is all real numbers except odd multiples of —.
2
At these x-values, the graph has vertical asymptotes.
π
Odd multiples of — are
2
values such as these:
• The domain of y = cot x is all real numbers except multiples of π.
At these x-values, the graph has vertical asymptotes.
π
π
±1 — = ± —
2
2
π
3π
±3 — = ± —
2
2
π
5π
±5 — = ± —
2
2
• The range of each function is all real numbers. So, the functions do not have
maximum or minimum values, and the graphs do not have an amplitude.
⋅
⋅
⋅
498
Chapter 9
hsnb_alg2_pe_0905.indd 498
• The period of each graph is π.
• The x-intercepts for y = tan x occur when x = 0, ±π, ±2π, ±3π, . . ..
π 3π 5π 7π
• The x-intercepts for y = cot x occur when x = ± —, ± —, ± —, ± —, . . ..
2
2
2
2
Trigonometric Ratios and Functions
2/5/15 1:51 PM
Graphing Tangent and Cotangent Functions
The graphs of y = a tan bx and y = a cot bx represent transformations of their parent
functions. The value of a indicates a vertical stretch (a > 1) or a vertical shrink
(0 < a < 1). The value of b indicates a horizontal stretch (0 < b < 1) or a horizontal
shrink (b > 1) and changes the period of the graph.
Core Concept
Period and Vertical Asymptotes of y = a tan bx and y = a cot bx
The period and vertical asymptotes of the graphs of y = a tan bx and y = a cot bx,
where a and b are nonzero real numbers, are as follows.
π
• The period of the graph of each function is —.
∣b∣
π
• The vertical asymptotes for y = a tan bx are at odd multiples of —.
2∣ b ∣
π
• The vertical asymptotes for y = a cot bx are at multiples of —.
∣b∣
Each graph below shows five key x-values that you can use to sketch the graphs of
y = a tan bx and y = a cot bx for a > 0 and b > 0. These are the x-intercept, the
x-values where the asymptotes occur, and the x-values halfway between the x-intercept
and the asymptotes. At each halfway point, the value of the function is either a or −a.
y
y
a
−
π
2b
a
π
4b
π
2b
π
4b
x
y = a tan bx
π
2b
π
b
x
y = a cot bx
Graphing a Tangent Function
Graph one period of g(x) = 2 tan 3x. Describe the graph of g as a transformation of the
graph of f (x) = tan x.
y
SOLUTION
4
π
−
6
π
12
−4
π
6
x
The function is of the form g(x) = a tan bx where a = 2 and b = 3. So, the period is
π
π
— = —.
∣b∣ 3
Intercept: (0, 0)
π
π
π
x = −— = −—, or x = −—
2(3)
6
2∣ b ∣
π
π
π
Asymptotes: x = — = —, or x = —;
2∣ b ∣ 2(3)
6
Halfway points:
π
π
, 2 ) = ( , 2 );
( 4bπ , a ) = ( 4(3)
12
—
—
—
π
π
, −2 ) = ( − , −2 )
( −4bπ , −a ) = ( −4(3)
12
—
—
—
The graph of g is a vertical stretch by a factor of 2 and a horizontal shrink by a
factor of —13 of the graph of f.
Section 9.5
hsnb_alg2_pe_0905.indd 499
Graphing Other Trigonometric Functions
499
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Graphing a Cotangent Function
Graph one period of g(x) = cot —12 x. Describe the graph of g as a transformation of the
graph of f (x) = cot x.
SOLUTION
The function is of the form g(x) = a cot bx where a = 1 and b = —12 . So, the period is
π
π
— = — = 2π.
∣ b ∣ —1
y
2
2
x
π
2
−2
π
2π
π
π
Intercept: —, 0 = —, 0 = (π, 0)
2b
2 —12
(
(() )
)
π
π
Asymptotes: x = 0; x = — = —, or x = 2π
∣ b ∣ —1
2
(() )
3π
π
π
π
3π
3π
Halfway points: —, a = —, 1 = —, 1 ; —, −a = —, −1 = —, −1
4b
2
4b
2
1
1
4 —2
4 —2
(
)
( )(
(() )
)
(
)
The graph of g is a horizontal stretch by a factor of 2 of the graph of f.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Graph one period of the function. Describe the graph of g as a transformation of
the graph of its parent function.
STUDY TIP
1
Because sec x = —,
cos x
sec x is undefined for
x-values at which
cos x = 0. The graph of
y = sec x has vertical
asymptotes at these
x-values. You can use
similar reasoning to
understand the vertical
asymptotes of the graph
of y = csc x.
1
1. g(x) = tan 2x
2. g(x) = —3 cot x
4. g(x) = 5 tan πx
3. g(x) = 2 cot 4x
Graphing Secant and Cosecant Functions
The graphs of secant and cosecant functions are related to the graphs of the parent
functions y = sec x and y = csc x, which are shown below.
3
y
y
2
y = sec x
π
−
2
π
2
5π x
2
y = cos x
−2π
y = sin x
−π
π
x
y = csc x
period: 2π
Core Concept
period: 2π
Characteristics of y = sec x and y = csc x
The functions y = sec x and y = csc x have the following characteristics.
π
• The domain of y = sec x is all real numbers except odd multiples of —.
2
At these x-values, the graph has vertical asymptotes.
• The domain of y = csc x is all real numbers except multiples of π.
At these x-values, the graph has vertical asymptotes.
• The range of each function is y ≤ −1 and y ≥ 1. So, the graphs do not have
an amplitude.
• The period of each graph is 2π.
500
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hsnb_alg2_pe_0905.indd 500
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To graph y = a sec bx or y = a csc bx, first graph the function y = a cos bx or
y = a sin bx, respectively. Then use the asymptotes and several points to sketch a
graph of the function. Notice that the value of b represents a horizontal stretch or
2π
1
shrink by a factor of —, so the period of y = a sec bx and y = a csc bx is —.
∣b∣
b
Graphing a Secant Function
Graph one period of g(x) = 2 sec x. Describe the graph of g as a transformation of the
graph of f (x) = sec x.
SOLUTION
Step 1 Graph the function y = 2 cos x.
2π
The period is — = 2π.
1
y
y = 2 sec x
3
y = 2 cos x
Step 2 Graph asymptotes of g. Because the
asymptotes of g occur when 2 cos x = 0,
π
π
3π
graph x = −—, x = —, and x = —.
2
2
2
π
2
π
x
−3
Step 3 Plot points on g, such as (0, 2) and
(π, −2). Then use the asymptotes to
sketch the curve.
The graph of g is a vertical stretch by a factor of 2 of the graph of f.
Graphing a Cosecant Function
LOOKING FOR
A PATTERN
In Examples 3 and 4,
notice that the plotted
points are on both
graphs. Also, these
points represent a local
maximum on one graph
and a local minimum on
the other graph.
1
Graph one period of g(x) = — csc πx. Describe the graph of g as a transformation of
2
the graph of f (x) = csc x.
SOLUTION
1
2π
Step 1 Graph the function y = — sin πx. The period is — = 2.
2
π
Step 2 Graph asymptotes of g. Because the
1
asymptotes of g occur when — sin πx = 0,
2
graph x = 0, x = 1, and x = 2.
y
1
( )
1 1
Step 3 Plot points on g, such as —, — and
2 2
3 1
—, −— . Then use the asymptotes to
2 2
sketch the curve.
(
)
1
x
2
y = 1 sin π x
2
y = 1 csc π x
2
1
The graph of g is a vertical shrink by a factor of — and a horizontal shrink by
2
1
a factor of — of the graph of f.
π
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Graph one period of the function. Describe the graph of g as a transformation of
the graph of its parent function.
5. g(x) = csc 3x
Section 9.5
hsnb_alg2_pe_0905.indd 501
1
6. g(x) = —2 sec x
7. g(x) = 2 csc 2x
8. g(x) = 2 sec πx
Graphing Other Trigonometric Functions
501
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9.5
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. WRITING Explain why the graphs of the tangent, cotangent, secant, and cosecant functions do not have
an amplitude.
2. COMPLETE THE SENTENCE The _______ and _______ functions are undefined for x-values at which sin x = 0.
3. COMPLETE THE SENTENCE The period of the function y = sec x is _____, and the period of y = cot x is _____.
4. WRITING Explain how to graph a function of the form y = a sec bx.
Monitoring Progress and Modeling with Mathematics
In Exercises 5–12, graph one period of the function.
Describe the graph of g as a transformation of the graph
of its parent function. (See Examples 1 and 2.)
16. USING EQUATIONS Which of the following are
asymptotes of the graph of y = 3 tan 4x?
5. g(x) = 2 tan x
6. g(x) = 3 tan x
π
A x=—
○
8
π
B x=—
○
4
7. g(x) = cot 3x
8. g(x) = cot 2x
C x=0
○
5π
D x = −—
○
8
1
1
9. g(x) = 3 cot —4 x
10. g(x) = 4 cot —2 x
1
In Exercises 17–24, graph one period of the function.
Describe the graph of g as a transformation of the graph
of its parent function. (See Examples 3 and 4.)
1
11. g(x) = —2 tan πx
12. g(x) = —3 tan 2πx
13. ERROR ANALYSIS Describe and correct the error in
finding the period of the function y = cot 3x.
✗
2π 2π
Period: — = —
∣b∣ 3
18. g(x) = 2 csc x
19. g(x) = sec 4x
20. g(x) = sec 3x
1
2
π
2
graph to write a function of the form y = a tan bx.
25.
−
graph each function.
y
y = 4 sin 3x
502
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hsnb_alg2_pe_0905.indd 502
π x
2
27.
−
π
2
π x
2
−1
−4
π
6
π
2
28.
y
y
2
x
x
−4
1
π
2
4
π
2
y
b. f (x) = 4 csc 3x
y = 3 cos 2x
π
4
26.
y
12
15. ANALYZING RELATIONSHIPS Use the given graph to
y
π
4
24. g(x) = csc — x
ATTENDING TO PRECISION In Exercises 25 –28, use the
A vertical stretch by a factor of 5 and
a horizontal shrink by a factor of —12.
4
22. g(x) = — sec 2πx
23. g(x) = csc — x
in describing the transformation of f (x) = tan x
represented by g(x) = 2 tan 5x.
a. f (x) = 3 sec 2x
1
4
21. g(x) = — sec πx
14. ERROR ANALYSIS Describe and correct the error
✗
17. g(x) = 3 csc x
−
1
2
5
1
2
x
π
−
4
π
4
x
Trigonometric Ratios and Functions
2/5/15 1:51 PM
40. f (x) = 4 csc x; vertical stretch by a factor of 2 and a
USING STRUCTURE In Exercises 29–34, match the
reflection in the x-axis
equation with the correct graph. Explain your
reasoning.
41. MULTIPLE REPRESENTATIONS Which function has
29. g(x) = 4 tan x
30. g(x) = 4 cot x
31. g(x) = 4 csc πx
32. g(x) = 4 sec πx
33. g(x) = sec 2x
34. g(x) = csc 2x
a greater local maximum value? Which has a greater
local minimum value? Explain.
1
A. f (x) = —4 csc πx
B.
y
4
A.
B.
y
y
1
π
2
π
−
2
x
π x
2
42. ANALYZING RELATIONSHIPS Order the functions
D.
y
from the least average rate of change to the greatest
π
π
average rate of change over the interval −— < x < —.
4
4
y
A.
4
4
π
2
E.
−4
F.
2
π
−
2
π x
2
π
−
2
π x
2
y
C.
x
D.
y
2
π
4
y
x
1
2
1
−1
B.
y
2
π x
y
π
−
4
x
−8
−1
C.
π
4
−4
4
y
2
2
x
1
π
−
2
−4
π
−
2
π x
2
π x
2
35. WRITING Explain why there is more than one tangent
function whose graph passes through the origin and
has asymptotes at x = −π and x = π.
36. USING EQUATIONS Graph one period of each
function. Describe the transformation of the graph of
its parent function.
a. g(x) = sec x + 3
b. g(x) = csc x − 2
c. g(x) = cot(x − π)
d. g(x) = −tan x
43. REASONING You are standing on a bridge 140 feet
above the ground. You look down at a car traveling
away from the underpass. The distance d (in feet) the
car is from the base of the bridge can be modeled by
d = 140 tan θ. Graph the function. Describe what
happens to θ as d increases.
θ
140 ft
WRITING EQUATIONS In Exercises 37– 40, write a rule
for g that represents the indicated transformation of the
graph of f.
π
37. f (x) = cot 2x; translation 3 units up and — units left
2
38. f (x) = 2 tan x; translation π units right, followed by
a horizontal shrink by a factor of —13
39. f (x) = 5 sec (x − π); translation 2 units down,
followed by a reflection in the x-axis
Section 9.5
hsnb_alg2_pe_0905.indd 503
d
44. USING TOOLS You use a video camera to pan up the
Statue of Liberty. The height h (in feet) of the part of
the Statue of Liberty that can be seen through your
video camera after time t (in seconds) can be modeled
π
by h = 100 tan — t. Graph the function using a
36
graphing calculator. What viewing window did you
use? Explain.
Graphing Other Trigonometric Functions
503
2/5/15 1:51 PM
45. MODELING WITH MATHEMATICS You are standing
48. HOW DO YOU SEE IT? Use the graph to answer
120 feet from the base of a 260-foot building. You
watch your friend go down the side of the building in
a glass elevator.
each question.
y
your friend
2
d
−3
−1
1
3 x
260 − d
θ
you
120 ft
a. What is the period of the graph?
Not drawn to scale
b. What is the range of the function?
a. Write an equation that gives the distance d (in
feet) your friend is from the top of the building as
a function of the angle of elevation θ.
b. Graph the function found in part (a). Explain how
the graph relates to this situation.
c. Is the function of the form f (x) = a csc bx or
f (x) = a sec bx? Explain.
49. ABSTRACT REASONING Rewrite a sec bx in terms
of cos bx. Use your results to explain the relationship
between the local maximums and minimums of the
cosine and secant functions.
46. MODELING WITH MATHEMATICS You are standing
300 feet from the base of a 200-foot cliff. Your friend
is rappelling down the cliff.
a. Write an equation that gives the distance d
(in feet) your friend is from the top of the cliff
as a function of the angle of elevation θ.
50. THOUGHT PROVOKING A trigonometric equation
that is true for all values of the variable for which
both sides of the equation are defined is called a
trigonometric identity. Use a graphing calculator to
graph the function
b. Graph the function found
in part (a).
c. Use a graphing calculator
to determine the angle of
elevation when your friend
has rappelled halfway
down the cliff.
x
1
x
y = — tan — + cot — .
2
2
2
(
)
Use your graph to write a trigonometric identity
involving this function. Explain your reasoning.
47. MAKING AN ARGUMENT Your friend states that it
51. CRITICAL THINKING Find a tangent function whose
is not possible to write a cosecant function that has
the same graph as y = sec x. Is your friend correct?
Explain your reasoning.
graph intersects the graph of y = 2 + 2 sin x only at
minimum points of the sine function.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Write a cubic function whose graph passes through the given points. (Section 4.9)
52. (−1, 0), (1, 0), (3, 0), (0, 3)
53. (−2, 0), (1, 0), (3, 0), (0, −6)
54. (−1, 0), (2, 0), (3, 0), (1, −2)
55. (−3, 0), (−1, 0), (3, 0), (−2, 1)
Find the amplitude and period of the graph of the function. (Section 9.4)
56.
57.
y
58.
y
6
2
5
π
−5
504
Chapter 9
hsnb_alg2_pe_0905.indd 504
2π
π
2
x
y
π
2π
x
6π
x
−2
−6
Trigonometric Ratios and Functions
2/5/15 1:51 PM
9.6
Modeling with Trigonometric
Functions
Essential Question
What are the characteristics of the real-life
problems that can be modeled by trigonometric functions?
Modeling Electric Currents
MODELING WITH
MATHEMATICS
Work with a partner. Find a sine function that models the electric current shown
in each oscilloscope screen. State the amplitude and period of the graph.
a.
To be proficient in math,
you need to apply the
mathematics you know
to solve problems arising
in everyday life.
15
10
10
5
5
0
0
-5
-5
-10
-10
-15
-15
1
2
3
4
5
6
7
8
9
-20
0
10
d.
20
15
10
10
5
5
0
0
-5
-5
-10
-10
-15
-15
1
2
3
4
5
6
7
8
9
-20
0
10
f.
20
15
10
10
5
5
0
0
-5
-5
-10
-10
-15
-15
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
20
15
-20
0
1
20
15
-20
0
e.
20
15
-20
0
c.
b.
20
-20
0
Communicate Your Answer
2. What are the characteristics of the real-life problems that can be modeled by
trigonometric functions?
3. Use the Internet or some other reference to find examples of real-life situations
that can be modeled by trigonometric functions.
Section 9.6
hsnb_alg2_pe_0906.indd 505
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505
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What You Will Learn
9.6 Lesson
Interpret and use frequency.
Write trigonometric functions.
Core Vocabul
Vocabulary
larry
Use technology to find trigonometric models.
frequency, p. 506
sinusoid, p. 507
Frequency
Previous
amplitude
period
midline
The periodic nature of trigonometric functions makes them useful for modeling
oscillating motions or repeating patterns that occur in real life. Some examples are
sound waves, the motion of a pendulum, and seasons of the year. In such applications,
the reciprocal of the period is called the frequency, which gives the number of cycles
per unit of time.
Using Frequency
A sound consisting of a single frequency is called a pure tone. An audiometer
produces pure tones to test a person’s auditory functions. An audiometer produces
a pure tone with a frequency f of 2000 hertz (cycles per second). The maximum
pressure P produced from the pure tone is 2 millipascals. Write and graph a sine
model that gives the pressure P as a function of the time t (in seconds).
SOLUTION
Step 1 Find the values of a and b in the model P = a sin bt. The maximum pressure
is 2, so a = 2. Use the frequency f to find b.
1
frequency = —
period
Write relationship involving frequency and period.
b
2000 = —
2π
Substitute.
4000π = b
Multiply each side by 2π.
The pressure P as a function of time t is given by P = 2 sin 4000π t.
Step 2 Graph the model. The amplitude is a = 2 and the period is
P
2
1
f
1
2000
— = —.
1
8000
−2
t
The key points are:
(
⋅
) (
)(
1
1
1
1
Intercepts: (0, 0); — —, 0 = —, 0 ; —, 0
2 2000
4000
2000
)
(⋅ ) ( )
3
1
3
, −2 ) = (
, −2 )
Minimum: ( ⋅
4 2000
8000
1
1
1
Maximum: — —, 2 = —, 2
4 2000
8000
—
—
—
The graph of P = 2 sin 4000π t is shown at the left.
506
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Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1. WHAT IF? In Example 1, how would the function change when the audiometer
produced a pure tone with a frequency of 1000 hertz?
Writing Trigonometric Functions
Graphs of sine and cosine functions are called sinusoids. One method to write a sine
or cosine function that models a sinusoid is to find the values of a, b, h, and k for
y = a sin b(x − h) + k
or
y = a cos b(x − h) + k
2π
where ∣ a ∣ is the amplitude, — is the period (b > 0), h is the horizontal shift, and k is
b
the vertical shift.
Writing a Trigonometric Function
Write a function for the sinusoid shown.
y
5
(π8 , 5(
3
x
π
( 38π , −1(
SOLUTION
Step 1 Find the maximum and minimum values. From the graph, the maximum
value is 5 and the minimum value is −1.
Step 2 Identify the vertical shift, k. The value of k is the mean of the maximum and
minimum values.
(maximum value) + (minimum value) 5 + (−1) 4
k = ———— = — = — = 2
2
2
2
STUDY TIP
Because the graph repeats
π
every — units, the period
2
π
is —.
2
Step 3 Decide whether the graph should be modeled by a sine or cosine function.
Because the graph crosses the midline y = 2 on the y-axis, the graph is a sine
curve with no horizontal shift. So, h = 0.
Step 4 Find the amplitude and period. The period is
Check
π
2
2π
b
—=—
6
b = 4.
The amplitude is
(maximum value) − (minimum value) 5 − (−1) 6
∣ a ∣ = ————
= — = — = 3.
2
π
−
2
2π
−2
2
The graph is not a reflection, so a > 0. Therefore, a = 3.
The function is y = 3 sin 4x + 2. Check this by graphing the function on a
graphing calculator.
Section 9.6
hsnb_alg2_pe_0906.indd 507
2
Modeling with Trigonometric Functions
507
2/5/15 1:52 PM
Modeling Circular Motion
Two people swing jump ropes, as shown in the diagram. The highest point of the
middle of each rope is 75 inches above the ground, and the lowest point is 3 inches.
The rope makes 2 revolutions per second. Write a model for the height h (in inches) of
a rope as a function of the time t (in seconds) given that the rope is at its lowest point
when t = 0.
75 in. above ground
3 in. above ground
Not drawn to scale
SOLUTION
A rope oscillates between 3 inches and 75 inches above the ground. So, a sine
or cosine function may be an appropriate model for the height over time.
Step 1 Identify the maximum and minimum values. The maximum height of
a rope is 75 inches. The minimum height is 3 inches.
Step 2 Identify the vertical shift, k.
(maximum value) + (minimum value) 75 + 3
k = ———— = — = 39
2
2
Check
Use the table feature of a
graphing calculator to check
your model.
X
Y1
.25
.5
.75
1
1.25
1.5
3
75
3
75
3
75
3
Step 3 Decide whether the height should be modeled by a sine or cosine function.
When t = 0, the height is at its minimum. So, use a cosine function whose
graph is a reflection in the x-axis with no horizontal shift (h = 0).
Step 4 Find the amplitude and period.
(maximum value) − (minimum value) 75 − 3
The amplitude is ∣ a ∣ = ———— = — = 36.
2
2
2 revolutions
Because the graph is a reflection in the x-axis, a < 0. So, a = −36. Because
a rope is rotating at a rate of 2 revolutions per second, one revolution is
2π
completed in 0.5 second. So, the period is — = 0.5, and b = 4π.
b
X=0
A model for the height of a rope is h(t) = −36 cos 4πt + 39.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Write a function for the sinusoid.
2.
3
3.
y
(0, 2)
−1
−3
(π3 , −2(
(12 , 1(
y
1
2π
3
x
1
2
3
2
5
2
x
(32 , −3(
4. WHAT IF? Describe how the model in Example 3 changes when the lowest point
of a rope is 5 inches above the ground and the highest point is 70 inches above
the ground.
508
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hsnb_alg2_pe_0906.indd 508
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Using Technology to Find Trigonometric Models
Another way to model sinusoids is to use a graphing calculator that has a sinusoidal
regression feature.
Using Sinusoidal Regression
T table shows the numbers N of hours of daylight in Denver, Colorado, on the
The
115th day of each month, where t = 1 represents January. Write a model that gives N
aas a function of t and interpret the period of its graph.
t
1
2
3
4
5
6
N
9.68
10.75
11.93
13.27
14.38
14.98
t
7
8
9
10
11
12
N
14.70
13.73
12.45
11.17
9.98
9.38
SOLUTION
S
Step 1 Enter the data in a graphing
S
calculator.
L1
L2
1
2
3
4
5
6
7
L3
9.68
10.75
11.93
13.27
14.38
14.98
14.7
Step 2
20
1
------
0
L1(1)=1
Step 3 The scatter plot appears
sinusoidal. So, perform a
sinusoidal regression.
STUDY TIP
Notice that the sinusoidal
regression feature finds
a model of the form
y = a sin(bx + c) + d. This
2π
function has a period of —
b
because it can be written
c
as y = a sin b x + — + d.
b
(
)
Make a scatter plot.
13
0
Step 4 Graph the data and the model
in the same viewing window.
20
SinReg
y=a*sin(bx+c)+d
a=2.764734198
b=.5111635715
c=-1.591149599
d=12.13293913
0
13
0
The model appears to be a good fit. So, a model for the data is
2π
N = 2.76 sin(0.511t − 1.59) + 12.1. The period, — ≈ 12, makes sense
0.511
because there are 12 months in a year and you would expect this pattern to
continue in following years.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
5. The table shows the average daily temperature T (in degrees Fahrenheit) for
a city each month, where m = 1 represents January. Write a model that gives T
as a function of m and interpret the period of its graph.
m
1
2
3
4
5
6
7
8
9
10
11
12
T
29
32
39
48
59
68
74
72
65
54
45
35
Section 9.6
hsnb_alg2_pe_0906.indd 509
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509
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9.6
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE Graphs of sine and cosine functions are called __________.
2. WRITING Describe how to find the frequency of the function whose graph is shown.
y
0.1
1
12
x
Monitoring Progress and Modeling with Mathematics
In Exercises 3 –10, find the frequency of the function.
3. y = sin x
4. y = sin 3x
5. y = cos 4x + 2
6. y = −cos 2x
7. y = sin 3πx
8. y = cos —
In Exercises 13–16, write a function for the sinusoid.
(See Example 2.)
13.
1
2
9. y = — cos 0.75x − 8
y
(π4 , 3)
2
x
πx
4
−
3π
4
π
−
4
π
4
3π
4
10. y = 3 sin 0.2x + 6
11. MODELING WITH MATHEMATICS The lowest
14.
frequency of sounds that can be heard by humans
is 20 hertz. The maximum pressure P produced
from a sound with a frequency of 20 hertz is
0.02 millipascal. Write and graph a sine model that
gives the pressure P as a function of the time t
(in seconds). (See Example 1.)
12. MODELING WITH MATHEMATICS A middle-A tuning
fork vibrates with a frequency f of 440 hertz (cycles
per second). You strike a middle-A tuning fork with a
force that produces a maximum pressure of 5 pascals.
Write and graph a sine model that gives the pressure P
as a function of the time t (in seconds).
5π
4
7π
4
( 34π , −3)
y
6
π
−
2
(0, 5)
π
2
−2
(π4 , −5)
−6
15.
y
x
(2, 2)
2
2
4
x
6
(0, −2)
16.
y
−1
1
( 32 , −1(
4
x
−2
( 12 , −3(
510
Chapter 9
hsnb_alg2_pe_0906.indd 510
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17. ERROR ANALYSIS Describe and correct the error in
finding the amplitude of a sinusoid with a maximum
point at (2, 10) and a minimum point at (4, −6).
✗
(maximum value) + (minimum value)
∣ a ∣ = ———
2
USING TOOLS In Exercises 21 and 22, the time t is
measured in months, where t = 1 represents January.
Write a model that gives the average monthly high
temperature D as a function of t and interpret the
period of the graph. (See Example 4.)
21.
10 − 6
=—
2
=2
18. ERROR ANALYSIS Describe and correct the error
in finding the vertical shift of a sinusoid with a
maximum point at (3, −2) and a minimum point
at (7, −8).
✗
(maximum value) + (minimum value)
k = ———
2
7+3
=—
2
=5
22.
Air Temperatures in Apple Valley, CA
t
1
2
3
4
5
6
D
60
63
69
75
85
94
t
7
8
9
10
11
12
D
99
99
93
81
69
60
Water Temperatures at Miami Beach, FL
t
1
2
3
4
5
6
D
71
73
75
78
81
85
t
7
8
9
10
11
12
D
86
85
84
81
76
73
23. MODELING WITH MATHEMATICS A circuit has an
19. MODELING WITH MATHEMATICS One of the largest
sewing machines in the world has a flywheel (which
turns as the machine sews) that is 5 feet in diameter.
The highest point of the handle at the edge of the
flywheel is 9 feet above the ground, and the lowest
point is 4 feet. The wheel makes a complete turn
every 2 seconds. Write a model for the height h
(in feet) of the handle as a function of the time t
(in seconds) given that the handle is at its lowest point
when t = 0. (See Example 3.)
20. MODELING WITH MATHEMATICS The Great Laxey
Wheel, located on the Isle of Man, is the largest
working water wheel in the world. The highest
point of a bucket on the wheel is 70.5 feet above the
viewing platform, and the lowest point is 2 feet below
the viewing platform. The wheel makes a complete
turn every 24 seconds. Write a model for the height h
(in feet) of the bucket as a function of time t
(in seconds) given that the bucket is at its lowest
point when t = 0.
alternating voltage of 100 volts that peaks every
0.5 second. Write a sinusoidal model for the voltage V
as a function of the time t (in seconds).
( 18 , 100(
V
100
t
1
8
( 38 , −100(
24. MULTIPLE REPRESENTATIONS The graph shows the
average daily temperature of Lexington, Kentucky.
The average daily temperature of Louisville,
π
Kentucky, is modeled by y = −22 cos —t + 57,
6
where y is the temperature (in degrees Fahrenheit) and
t is the number of months since January 1. Which city
has the greater average daily temperature? Explain.
Temperature
(F°)
Daily Temperature in Lexington
T
80
(6, 76)
40
(0, 33)
0
0
2
4
6
8
10
t
Months since January 1
Section 9.6
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25. USING TOOLS The table shows the numbers of
28. HOW DO YOU SEE IT? What is the frequency of the
employees N (in thousands) at a sporting goods
company each year for 11 years. The time t is
measured in years, with t = 1 representing the
first year.
t
1
2
3
4
5
6
N
20.8
22.7
24.6
23.2
20
17.5
t
7
8
9
10
11
12
N
16.7
17.8
21
22
24.1
function whose graph is shown? Explain.
y
0.5
x
1
8
13
8
17
8
21
8
π
π
has a minimum at —, 3 and a maximum at —, 8 .
2
4
Write a sine function and a cosine function for the
sinusoid. Use a graphing calculator to verify that
your answers are correct.
( )
b. Predict the number of employees at the company
in the 12th year.
26. THOUGHT PROVOKING The figure shows a tangent
( )
30. MAKING AN ARGUMENT Your friend claims that a
line drawn to the graph of the function y = sin x. At
several points on the graph, draw a tangent line to
the graph and estimate its slope. Then plot the points
(x, m), where m is the slope of the tangent line. What
can you conclude?
function with a frequency of 2 has a greater period
than a function with a frequency of —12. Is your friend
correct? Explain your reasoning.
31. PROBLEM SOLVING The low tide at a port is 3.5 feet
y
and occurs at midnight. After 6 hours, the port is at
high tide, which is 16.5 feet.
1
−π
9
8
29. USING STRUCTURE During one cycle, a sinusoid
a. Use sinusoidal regression to find a model that
gives N as a function of t.
−2π
5
8
π
2π
x
The slope of the tangent
line at (0, 0) is 1.
high
h tid
ide: 16.5
5 ft
low
lo
w tiide
e: 3.5
5 ft
ft
27. REASONING Determine whether you would use a sine
or cosine function to model each sinusoid with the
y-intercept described. Explain your reasoning.
a. Write a sinusoidal model that gives the tide depth
d (in feet) as a function of the time t (in hours). Let
t = 0 represent midnight.
a. The y-intercept occurs at the maximum value of
the function.
b. Find all the times when low and high tides occur
in a 24-hour period.
b. The y-intercept occurs at the minimum value of
the function.
c. Explain how the graph of the function you wrote
in part (a) is related to a graph that shows the tide
depth d at the port t hours after 3:00 a.m.
c. The y-intercept occurs halfway between the
maximum and minimum values of the function.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Simplify the expression. (Section 5.2)
3
17
32. —
—
√2
33. —
—
√6 − 2
8
13
34. —
—
35. —
—
—
√3 + √11
38. log3 5x3
39. ln —
√10 + 3
Expand the logarithmic expression. (Section 6.5)
x
7
36. log8 —
512
Chapter 9
hsnb_alg2_pe_0906.indd 512
37. ln 2x
4x6
y
Trigonometric Ratios and Functions
2/5/15 1:52 PM
9.7
Using Trigonometric Identities
Essential Question
How can you verify a trigonometric
identity?
Writing a Trigonometric Identity
Work with a partner. In the figure, the point
(x, y) is on a circle of radius c with center at
the origin.
y
(x, y)
a. Write an equation that relates a, b, and c.
b. Write expressions for the sine and cosine
ratios of angle θ.
c
c. Use the results from parts (a) and (b) to
find the sum of sin2θ and cos2θ. What do
you observe?
θ
b
a
x
d. Complete the table to verify that the identity you wrote in part (c) is valid
for angles (of your choice) in each of the four quadrants.
θ
sin2 θ
cos2 θ
sin2 θ + cos2 θ
QI
QII
QIII
QIV
Writing Other Trigonometric Identities
REASONING
ABSTRACTLY
To be proficient in math,
you need to know and
flexibly use different
properties of operations
and objects.
Work with a partner. The trigonometric identity you derived in Exploration 1 is
called a Pythagorean identity. There are two other Pythagorean identities. To derive
them, recall the four relationships:
sin θ
tan θ = —
cos θ
cos θ
cot θ = —
sin θ
1
sec θ = —
cos θ
1
csc θ = —
sin θ
a. Divide each side of the Pythagorean identity you derived in Exploration 1
by cos2θ and simplify. What do you observe?
b. Divide each side of the Pythagorean identity you derived in Exploration 1
by sin2θ and simplify. What do you observe?
Communicate Your Answer
3. How can you verify a trigonometric identity?
4. Is sin θ = cos θ a trigonometric identity? Explain your reasoning.
5. Give some examples of trigonometric identities that are different than those in
Explorations 1 and 2.
Section 9.7
hsnb_alg2_pe_0907.indd 513
Using Trigonometric Identities
513
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9.7
Lesson
What You Will Learn
Use trigonometric identities to evaluate trigonometric functions and
simplify trigonometric expressions.
Core Vocabul
Vocabulary
larry
Verify trigonometric identities.
trigonometric identity, p. 514
Previous
unit circle
Using Trigonometric Identities
Recall that when an angle θ is in standard
position with its terminal side intersecting
the unit circle at (x, y), then x = cos θ and
y = sin θ. Because (x, y) is on a circle
centered at the origin with radius 1, it
follows that
STUDY TIP
Note that sin2 θ represents
(sin θ)2 and cos2 θ
represents (cos θ)2.
y
r=1
(cos θ, sin θ) = (x, y)
θ
x
x2 + y2 = 1
and
cos2 θ + sin2 θ = 1.
The equation cos2 θ + sin2 θ = 1 is true for any value of θ. A trigonometric equation
that is true for all values of the variable for which both sides of the equation are
defined is called a trigonometric identity. In Section 9.1, you used reciprocal
identities to find the values of the cosecant, secant, and cotangent functions. These
and other fundamental trigonometric identities are listed below.
Core Concept
Fundamental Trigonometric Identities
Reciprocal Identities
1
csc θ = —
sin θ
1
sec θ = —
cos θ
1
cot θ = —
tan θ
Tangent and Cotangent Identities
sin θ
tan θ = —
cos θ
cos θ
cot θ = —
sin θ
Pythagorean Identities
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
π
cos — − θ = sin θ
2
π
tan — − θ = cot θ
2
cos(−θ) = cos θ
tan(−θ) = −tan θ
Cofunction Identities
π
sin — − θ = cos θ
2
(
)
(
)
(
)
Negative Angle Identities
sin(−θ) = −sin θ
In this section, you will use trigonometric identities to do the following.
• Evaluate trigonometric functions.
• Simplify trigonometric expressions.
• Verify other trigonometric identities.
514
Chapter 9
hsnb_alg2_pe_0907.indd 514
Trigonometric Ratios and Functions
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Finding Trigonometric Values
4
π
Given that sin θ = — and — < θ < π, find the values of the other five trigonometric
5
2
functions of θ.
SOLUTION
Step 1 Find cos θ.
sin2 θ + cos2 θ = 1
Write Pythagorean identity.
2
( 45 ) + cos θ = 1
4
Substitute — for sin θ.
5
2
—
4 2
42
cos2 θ = 1 − —
Subtract — from each side.
5
5
9
cos2 θ = —
Simplify.
25
3
cos θ = ± —
Take square root of each side.
5
3
cos θ = −—
Because θ is in Quadrant II, cos θ is negative.
5
Step 2 Find the values of the other four trigonometric functions of θ using the values
of sin θ and cos θ.
4
3
−—
—
5
5
4
3
sin θ
cos θ
cot θ = — = — = −—
tan θ = — = — = −—
cos θ
3
sin θ
4
3
4
−—
—
5
5
()
()
1
5
1
csc θ = — = — = —
sin θ
4
4
—
5
1
5
1
sec θ = — = — = −—
cos θ
3
3
−—
5
Simplifying Trigonometric Expressions
π
Simplify (a) tan — − θ sin θ and (b) sec θ tan2 θ + sec θ.
2
(
)
SOLUTION
π
a. tan — − θ sin θ = cot θ sin θ
2
cos θ
= — (sin θ)
sin θ
= cos θ
(
)
Cofunction identity
( )
Cotangent identity
Simplify.
b. sec θ tan2 θ + sec θ = sec θ(sec2 θ − 1) + sec θ
Pythagorean identity
= sec3 θ − sec θ + sec θ
Distributive Property
= sec3 θ
Simplify.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1
6
trigonometric functions of θ.
π
2
1. Given that cos θ = — and 0 < θ < —, find the values of the other five
Simplify the expression.
2. sin x cot x sec x
3. cos θ − cos θ sin2 θ
Section 9.7
hsnb_alg2_pe_0907.indd 515
tan x csc x
sec x
4. —
Using Trigonometric Identities
515
2/5/15 1:54 PM
Verifying Trigonometric Identities
You can use the fundamental identities from this chapter to verify new trigonometric
identities. When verifying an identity, begin with the expression on one side. Use
algebra and trigonometric properties to manipulate the expression until it is identical
to the other side.
Verifying a Trigonometric Identity
sec2 θ − 1
Verify the identity —
= sin2 θ.
sec2 θ
SOLUTION
sec2 θ − 1
sec θ
sec2 θ
sec θ
1
sec θ
=—
−—
—
2
2
2
Write as separate fractions.
2
( )
1
=1− —
sec θ
Simplify.
= 1 − cos2 θ
Reciprocal identity
= sin2 θ
Pythagorean identity
Notice that verifying an identity is not the same as solving an equation. When
verifying an identity, you cannot assume that the two sides of the equation are equal
because you are trying to verify that they are equal. So, you cannot use any properties
of equality, such as adding the same quantity to each side of the equation.
Verifying a Trigonometric Identity
cos x
Verify the identity sec x + tan x = —.
1 − sin x
SOLUTION
LOOKING FOR
STRUCTURE
To verify the identity, you
must introduce 1 − sin x
into the denominator.
Multiply the numerator
and the denominator by
1 − sin x so you get an
equivalent expression.
1
sec x + tan x = — + tan x
cos x
Reciprocal identity
sin x
1
=—+—
cos x cos x
Tangent identity
1 + sin x
=—
cos x
Add fractions.
1 + sin x 1 − sin x
=— —
cos x
1 − sin x
1 − sin x
Multiply by —.
1 − sin x
1 − sin2 x
= ——
cos x(1 − sin x)
Simplify numerator.
cos2 x
= ——
cos x(1 − sin x)
Pythagorean identity
cos x
=—
1 − sin x
Simplify.
⋅
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Verify the identity.
516
Chapter 9
hsnb_alg2_pe_0907.indd 516
5. cot(−θ) = −cot θ
6. csc2 x(1 − sin2 x) = cot2 x
7. cos x csc x tan x = 1
8. (tan2 x + 1)(cos2 x − 1) = −tan2 x
Trigonometric Ratios and Functions
2/5/15 1:54 PM
9.7
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. WRITING Describe the difference between a trigonometric identity and a trigonometric equation.
2. WRITING Explain how to use trigonometric identities to determine whether sec(−θ) = sec θ or
sec(−θ) = −sec θ.
Monitoring Progress and Modeling with Mathematics
In Exercises 3 –10, find the values of the other five
trigonometric functions of θ. (See Example 1.)
π
2
1
3
3. sin θ = —, 0 < θ < —
ERROR ANALYSIS In Exercises 21 and 22, describe and
correct the error in simplifying the expression.
21.
✗
22.
✗
3π
2
7
10
4. sin θ = −—, π < θ < —
3 π
7 2
5. tan θ = −—, — < θ < π
2 π
5 2
6. cot θ = −—, — < θ < π
3π
2
5
6
1 − sin2 θ = 1 − (1 + cos2 θ )
= 1 − 1 − cos2 θ
= −cos2 θ
cos x 1
tan x csc x = — —
sin x sin x
cos x
=—
sin2 x
⋅
7. cos θ = −—, π < θ < —
9 3π
4 2
8. sec θ = —, — < θ < 2π
In Exercises 23–30, verify the identity. (See Examples 3
and 4.)
3π
2
9. cot θ = −3, — < θ < 2π
23. sin x csc x = 1
3π
2
5
3
24. tan θ csc θ cos θ = 1
( π2 )
π
sin( − x ) tan x = sin x
2
π
cos( − θ ) + 1
2
= 1 28.
10. csc θ = −—, π < θ < —
25. cos — − x cot x = cos x
In Exercises 11–20, simplify the expression.
(See Example 2.)
26.
12. cos θ (1 + tan2 θ)
11. sin x cot x
sin(−θ)
cos(−θ)
cos2 x
cot x
13. —
14. —
2
π
cos — − x
2
15. —
csc x
(
)
( π2 )
16. sin — − θ sec θ
csc2 x − cot2 x
sin(−x) cot x
cos2 x tan2(−x) − 1
cos x
17. ——
18. ——
2
π
cos — − θ
2
19. — + cos2 θ
csc θ
π
sec x sin x + cos — − x
2
20. ———
1 + sec x
(
)
(
)
—
—
27. ——
1 − sin(−θ)
1 + cos x
sin x
sin x
1 + cos x
29. — + — = 2 csc x
sin x
1 − cos(−x)
30. —— = csc x + cot x
31. USING STRUCTURE A function f is odd when
f (−x) = −f(x). A function f is even when
f (−x) = f (x). Which of the six trigonometric
functions are odd? Which are even? Justify your
answers using identities and graphs.
32. ANALYZING RELATIONSHIPS As the value of cos θ
increases, what happens to the value of sec θ? Explain
your reasoning.
Section 9.7
hsnb_alg2_pe_0907.indd 517
sin2(−x)
tan x
= cos2 x
—
2
Using Trigonometric Identities
517
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33. MAKING AN ARGUMENT Your friend simplifies
37. DRAWING CONCLUSIONS Static friction is the amount
an expression and obtains sec x tan x − sin x. You
simplify the same expression and obtain sin x tan2 x.
Are your answers equivalent? Justify your answer.
of force necessary to keep a stationary object on a
flat surface from moving. Suppose a book weighing
W pounds is lying on a ramp inclined at an angle θ.
The coefficient of static friction u for the book can be
found using the equation uW cos θ = W sin θ.
34. HOW DO YOU SEE IT? The figure shows the unit
circle and the angle θ.
a. Solve the equation for u and simplify the result.
a. Is sin θ positive or negative? cos θ? tan θ?
b. Use the equation from part (a) to determine what
happens to the value of u as the angle θ increases
from 0° to 90°.
b. In what quadrant does the terminal side of −θ lie?
c. Is sin(−θ) positive or negative? cos(−θ)?
tan(−θ)?
38. PROBLEM SOLVING When light traveling in a medium
(such as air) strikes the surface of a second medium
(such as water) at an angle θ1, the light begins to
travel at a different angle θ2. This change of direction
is defined by Snell’s law, n1 sin θ1 = n2 sin θ2, where
n1 and n2 are the indices of refraction for the two
mediums. Snell’s law can be derived from the equation
y
(x, y)
θ
x
n1
n2
√cot2 θ1 + 1
√cot2 θ2 + 1
——
— = ——
—.
35. MODELING WITH MATHEMATICS A vertical gnomon
(the part of a sundial that projects a shadow) has
height h. The length s of the shadow cast by the
gnomon when the angle of the Sun above the horizon
is θ can be modeled by the equation below. Show that
the equation below is equivalent to s = h cot θ.
h sin(90° − θ)
s =——
sin θ
h
s
air: n1
θ1
water: n2
θ2
a. Simplify the equation to derive Snell’s law.
b. What is the value of n1 when θ1 = 55°, θ2 = 35°,
and n2 = 2?
c. If θ1 = θ2, then what must be true about the
values of n1 and n2? Explain when this situation
would occur.
39. WRITING Explain how transformations of the
graph of the parent function f (x) = sin x support the
π
cofunction identity sin — − θ = cos θ.
2
θ
)
(
40. USING STRUCTURE Verify each identity.
a. ln∣ sec θ ∣ = −ln∣ cos θ ∣
36. THOUGHT PROVOKING Explain how you can use a
trigonometric identity to find all the values of x for
which sin x = cos x.
Maintaining Mathematical Proficiency
b. ln∣ tan θ ∣ = ln∣ sin θ ∣ − ln∣ cos θ ∣
Reviewing what you learned in previous grades and lessons
Find the value of x for the right triangle. (Section 9.1)
41.
42.
43.
13
11
x
7
30°
45°
x
518
Chapter 9
hsnb_alg2_pe_0907.indd 518
60°
x
Trigonometric Ratios and Functions
2/5/15 1:54 PM
9.8
Using Sum and Difference Formulas
Essential Question
How can you evaluate trigonometric
functions of the sum or difference of two angles?
Deriving a Difference Formula
Work with a partner.
a. Explain why the two triangles shown are congruent.
y
(cos a, sin a)
y
d
(cos(a − b), sin(a − b))
(cos b, sin b)
1
a
d
1
a−b
b
x
CONSTRUCTING
VIABLE ARGUMENTS
To be proficient in math,
you need to understand
and use stated assumptions,
definitions, and previously
established results.
(1, 0)
x
b. Use the Distance Formula to write an expression for d in the first unit circle.
c. Use the Distance Formula to write an expression for d in the second unit circle.
d. Write an equation that relates the expressions in parts (b) and (c). Then simplify
this equation to obtain a formula for cos(a − b).
Deriving a Sum Formula
Work with a partner. Use the difference formula you derived in Exploration 1 to write
a formula for cos(a + b) in terms of sine and cosine of a and b. Hint: Use the fact that
cos(a + b) = cos[a − (−b)].
Deriving Difference and Sum Formulas
Work with a partner. Use the formulas you derived in Explorations 1 and 2 to write
formulas for sin(a − b) and sin(a + b) in terms of sine and cosine of a and b. Hint:
Use the cofunction identities
π
π
sin — − a = cos a and cos — − a = sin a
2
2
(
)
(
)
and the fact that
π
cos — − a + b = sin(a − b) and sin(a + b) = sin[a − (−b)].
2
[(
) ]
Communicate Your Answer
4. How can you evaluate trigonometric functions of the sum or difference of
two angles?
5. a. Find the exact values of sin 75° and cos 75° using sum formulas. Explain
your reasoning.
b. Find the exact values of sin 75° and cos 75° using difference formulas.
Compare your answers to those in part (a).
Section 9.8
hsnb_alg2_pe_0908.indd 519
Using Sum and Difference Formulas
519
2/5/15 1:54 PM
9.8 Lesson
What You Will Learn
Use sum and difference formulas to evaluate and simplify trigonometric
expressions.
Core Vocabul
Vocabulary
larry
Use sum and difference formulas to solve trigonometric equations and
rewrite real-life formulas.
Previous
ratio
Using Sum and Difference Formulas
In this lesson, you will study formulas that allow you to evaluate trigonometric
functions of the sum or difference of two angles.
Core Concept
Sum and Difference Formulas
Difference Formulas
Sum Formulas
sin(a + b) = sin a cos b + cos a sin b
sin(a − b) = sin a cos b − cos a sin b
cos(a + b) = cos a cos b − sin a sin b
cos(a − b) = cos a cos b + sin a sin b
tan a + tan b
tan(a + b) = ——
1 − tan a tan b
tan a − tan b
tan(a − b) = ——
1 + tan a tan b
In general, sin(a + b) ≠ sin a + sin b. Similar statements can be made for the other
trigonometric functions of sums and differences.
Evaluating Trigonometric Expressions
7π
Find the exact value of (a) sin 15° and (b) tan —.
12
SOLUTION
a. sin 15° = sin(60° − 45°)
Check
Substitute 60° − 45° for 15°.
= sin 60° cos 45° − cos 60° sin 45°
sin(15˚)
.2588190451
—
—
( ) ( )
√3 √2
1 √2
=— — −— —
2 2
2 2
( (6)- (2))/4
.2588190451
—
Difference formula for sine
—
Evaluate.
—
√6 − √2
=—
4
Simplify.
—
—
√6 − √2
The exact value of sin 15° is —. Check this with a calculator.
4
π π
7π
b. tan — = tan — + —
12
3
4
π
π
tan — + tan —
3
4
= ——
π
π
1 − tan — tan —
3
4
(
Check
tan(7π/12)
-3.732050808
-2- (3)
-3.732050808
)
π π 7π
Substitute — + — for —.
12
3
4
Sum formula for tangent
—
√3 + 1
=—
—
1 − √3 1
⋅
Evaluate.
= −2 − √ 3
Simplify.
—
—
7π
The exact value of tan — is −2 − √ 3 . Check this with a calculator.
12
520
Chapter 9
hsnb_alg2_pe_0908.indd 520
Trigonometric Ratios and Functions
2/5/15 1:54 PM
Using a Difference Formula
ANOTHER WAY
You can also use a
Pythagorean identity and
quadrant signs to find
sin a and cos b.
4
5
3π
Find cos(a − b) given that cos a = −— with π < a < — and sin b = — with
5
2
13
π
0 < b < —.
2
SOLUTION
Step 1 Find sin a and cos b.
4
Because cos a = −— and a is in
5
3
Quadrant III, sin a = −—, as
5
shown in the figure.
5
Because sin b = — and b is in
13
12
Quadrant I, cos b = —, as shown
13
in the figure.
y
y
4
52 − 42 = 3
13
b
a
5
x
x
132 − 52 = 12
5
Step 2 Use the difference formula for cosine to find cos(a − b).
cos(a − b) = cos a cos b + sin a sin b
( ) ( )( )
Difference formula for cosine
4 12
3 5
= −— — + −— —
5 13
5 13
Evaluate.
63
= −—
65
Simplify.
63
The value of cos(a − b) is −—.
65
Simplifying an Expression
Simplify the expression cos(x + π).
SOLUTION
cos(x + π) = cos x cos π − sin x sin π
Sum formula for cosine
= (cos x)(−1) − (sin x)(0)
Evaluate.
= −cos x
Simplify.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Find the exact value of the expression.
π
12
π
24
8
5. Find sin(a − b) given that sin a = — with 0 < a < — and cos b = −—
17
2
25
3π
with π < b < —.
2
1. sin 105°
5π
12
2. cos 15°
3. tan —
4. cos —
Simplify the expression.
6. sin(x + π)
7. cos(x − 2π)
Section 9.8
hsnb_alg2_pe_0908.indd 521
8. tan(x − π)
Using Sum and Difference Formulas
521
2/5/15 1:54 PM
Solving Equations and Rewriting Formulas
Solving a Trigonometric Equation
π
π
Solve sin x + — + sin x − — = 1 for 0 ≤ x < 2π.
3
3
(
)
(
)
SOLUTION
ANOTHER WAY
You can also solve the
equation by using a
graphing calculator. First,
graph each side of the
original equation. Then
use the intersect feature
to find the x-value(s)
where the expressions
are equal.
π
π
sin x + — + sin x − — = 1
3
3
π
π
π
π
sin x cos — + cos x sin — + sin x cos — − cos x sin — = 1
3
3 —
3
3
—
√3
√3
1
1
—sin x + —cos x + —sin x − —cos x = 1
2
2
2
2
sin x = 1
(
)
(
)
Write equation.
Use formulas.
Evaluate.
Simplify.
π
In the interval 0 ≤ x < 2π, the solution is x = —.
2
Rewriting a Real-Life Formula
air
α
θ
t
h
lig
prism
The index of refraction of a transparent material is the ratio of the speed of light in a
vacuum to the speed of light in the material. A triangular prism, like the one shown,
can be used to measure the index of refraction using the formula
θ α
sin — + —
2 2
n = —.
θ
sin —
2
(
)
—
√3 1
θ
For α = 60°, show that the formula can be rewritten as n = — + — cot —.
2
2
2
SOLUTION
)
θ
sin — + 30°
2
n = ——
θ
sin —
2
θ
θ
sin — cos 30° + cos — sin 30°
2
2
= ———
θ
sin —
2
—
θ √3
θ 1
sin — — + cos — —
2 2
2 2
= ———
θ
sin —
2
—
√3
θ 1
θ
—sin —
— cos —
2
2 2
2
=—+—
θ
θ
sin —
sin —
2
2
—
√3 1
θ
= — + — cot —
2
2
2
(
( )( ) ( )( )
Monitoring Progress
( π4 )
(
α 60°
Write formula with — = — = 30°.
2
2
Sum formula for sine
Evaluate.
Write as separate fractions.
Simplify.
Help in English and Spanish at BigIdeasMath.com
π
4
)
9. Solve sin — − x − sin x + — = 1 for 0 ≤ x < 2π.
522
Chapter 9
hsnb_alg2_pe_0908.indd 522
Trigonometric Ratios and Functions
2/5/15 1:55 PM
9.8
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE Write the expression cos 130° cos 40° − sin 130° sin 40° as the cosine
of an angle.
2. WRITING Explain how to evaluate tan 75° using either the sum or difference formula for tangent.
Monitoring Progress and Modeling with Mathematics
In Exercises 3–10, find the exact value of the expression.
(See Example 1.)
3. tan(−15°)
24.
4. tan 195°
23π
12
5. sin —
17π
12
11. sin(a + b)
12. sin(a − b)
13. cos(a − b)
14. cos(a + b)
15. tan(a + b)
16. tan(a − b)
for 0 ≤ x < 2π?
π
A —
○
3
2π
C —
○
3
π
18. cos x − —
2
19. cos(x + 2π)
3π
2
21. sin x − —
(
π
2
22. tan x + —
)
π
tan x + tan —
π
4
tan x + — = ——
π
4
1 + tan x tan —
4
tan x + 1
=—
1 + tan x
)
29.
30.
π
28. tan( x − ) = 0
( π2 ) 12
4
π
π
cos( x + ) − cos( x − ) = 1
6
6
π
π
sin( x + ) + sin( x − ) = 0
4
4
—
—
—
—
—
31. tan(x + π) − tan(π − x) = 0
32. sin(x + π) + cos(x + π) = 0
33. USING EQUATIONS Derive the cofunction identity
π
sin — − θ = cos θ using the difference formula
2
for sine.
(
=1
Section 9.8
hsnb_alg2_pe_0908.indd 523
7π
D —
○
4
27. sin x + — = —
)
correct the error in simplifying the expression.
3π
B —
○
4
In Exercises 27– 32, solve the equation for 0 ≤ x < 2π.
(See Example 4.)
ERROR ANALYSIS In Exercises 23 and 24, describe and
(
5π
D —
○
6
for 0 ≤ x < 2π ?
π
A —
○
4
5π
C —
○
4
20. tan(x − 2π)
)
π
B —
○
6
26. What are the solutions of the equation tan x + 1 = 0
In Exercises 17–22, simplify the expression.
(See Example 3.)
(
—
25. What are the solutions of the equation 2 sin x − 1 = 0
In Exercises 11–16, evaluate the expression given
π
15
4
that cos a = — with 0 < a < — and sin b = −— with
5
2
17
3π
< b < 2π. (See Example 2.)
—
2
✗
—
√2
√2
= — cos x − — sin x
2
2
—
( )
17. tan(x + π)
)
√2
= — (cos x − sin x)
2
11π
8. cos —
12
7π
10. sin −—
12
9. tan —
23.
(
π
π
π
sin x − — = sin — cos x − cos — sin x
4
4
4
6. sin(−165°)
7. cos 105°
(
✗
)
Using Sum and Difference Formulas
523
2/5/15 1:55 PM
34. MAKING AN ARGUMENT Your friend claims it is
38. HOW DO YOU SEE IT? Explain how to use the figure
π
π
to solve the equation sin x + — − sin — − x = 0
4
4
for 0 ≤ x < 2π.
(
possible to use the difference formula for tangent to
π
derive the cofunction identity tan — − θ = cot θ. Is
2
your friend correct? Explain your reasoning.
(
)
y
(
f(x) = sin x +
35. MODELING WITH MATHEMATICS A photographer
is at a height h taking aerial photographs with a
35-millimeter camera. The ratio of the image length
WQ to the length NA of the actual object is given by
the formula
WQ
NA
35 tan(θ − t) + 35 tan t
h tan θ
h
— = ——
(
π
4
)
(
x
π
−1
camera
θ
)
g(x) = sin
Q
2π
( π4 − x(
t
W
39. MATHEMATICAL CONNECTIONS The figure shows the
N
acute angle of intersection, θ2 − θ1, of two lines with
slopes m1 and m2.
A
where θ is the angle between the vertical line
perpendicular to the ground and the line from the
camera to point A and t is the tilt angle of the film.
When t = 45°, show that the formula can be rewritten
70
WQ
as — = ——. (See Example 5.)
NA
h(1 + tan θ)
y
y = m1x + b1
y = m2 x + b2
θ 2 − θ1
36. MODELING WITH MATHEMATICS When a wave
travels through a taut string, the displacement y of
each point on the string depends on the time t and the
point’s position x. The equation of a standing wave
can be obtained by adding the displacements of two
waves traveling in opposite directions. Suppose a
standing wave can be modeled by the formula
2πt 2πx
2πt 2πx
y = A cos — − — + A cos — + — .
3
5
3
5
When t = 1, show that the formula can be rewritten as
2πx
y = −A cos — .
5
(
)
(
θ1
x
a. Use the difference formula for tangent to write an
equation for tan (θ2 − θ1) in terms of m1 and m2.
)
37. MODELING WITH MATHEMATICS The busy signal on
a touch-tone phone is a combination of two tones with
frequencies of 480 hertz and 620 hertz. The individual
tones can be modeled by the equations:
θ2
b. Use the equation from part (a) to find the acute
angle of intersection of the
lines y = x − 1 and
—
4 − √3
1
y= —
x+—
—
—.
√3 − 2
2 − √3
(
)
40. THOUGHT PROVOKING Rewrite each function. Justify
your answers.
a. Write sin 3x as a function of sin x.
480 hertz: y1 = cos 960πt
b. Write cos 3x as a function of cos x.
620 hertz: y2 = cos 1240πt
c. Write tan 3x as a function of tan x.
The sound of the busy signal can be modeled by
y1 + y2. Show that y1 + y2 = 2 cos 1100πt cos 140πt.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Solve the equation. Check your solution(s). (Section 7.5)
9
x−2
7
2
41. 1 − — = −—
524
Chapter 9
hsnb_alg2_pe_0908.indd 524
12
x
3
4
8
x
42. — + — = —
2x − 3
x+1
10
x −1
+5
43. — = —
2
Trigonometric Ratios and Functions
2/5/15 1:55 PM
9.5–9.8
What Did You Learn?
Core Vocabulary
frequency, p. 506
sinusoid, p. 507
trigonometric identity, p. 514
Core Concepts
Section 9.5
Characteristics of y = tan x and y = cot x, p. 498
Period and Vertical Asymptotes of y = a tan bx and y = a cot bx, p. 499
Characteristics of y = sec x and y = csc x, p. 500
Section 9.6
Frequency, p. 506
Writing Trigonometric Functions, p. 507
Using Technology to Find Trigonometric Models, p. 509
Section 9.7
Fundamental Trigonometric Identities, p. 514
Section 9.8
Sum and Difference Formulas, p. 520
Trigonometric Equations and Real-Life Formulas, p. 522
Mathematical Practices
1.
Explain why the relationship between θ and d makes sense in the context of the situation
in Exercise 43 on page 503.
2.
How can you use definitions to relate the slope of a line with the tangent off an ang
angle
glee iin
n
Exercise 39 on page 524?
Performance Task
Lightening the Load
You need to move a heavy table across the room. What is the easiestt
way to move it? Should you push it? Should you tie a rope around one
nee
leg of the table and pull it? How can trigonometry help you make the
e
right decision?
To explore the answers to these questions and more, go to
BigIdeasMath.com.
525
9
Chapter Review
9.1
Dynamic Solutions available at BigIdeasMath.com
Right Triangle Trigonometry (pp. 461−468)
Evaluate the six trigonometric functions of the angle θ.
From the Pythagorean Theorem, the length of the hypotenuse is
6
—
hyp. = √62 + 82
—
θ
= √100
8
= 10.
Using adj. = 8, opp. = 6, and hyp. = 10, the values of the six trigonometric functions of θ are:
6
3
opp.
sin θ = — = — = —
hyp. 10 5
8
4
adj.
cos θ = — = — = —
hyp. 10 5
opp. 6 3
tan θ = — = — = —
adj.
8 4
hyp. 10 5
csc θ = — = — = —
opp.
6
3
hyp. 10 5
sec θ = — = — = —
adj.
8
4
8 4
adj.
cot θ = — = — = —
opp. 6 3
6
1. In a right triangle, θ is an acute angle and cos θ = —
. Evaluate the other five trigonometric
11
functions of θ.
2. The shadow of a tree measures 25 feet from its base. The angle of elevation to the Sun is 31°.
How tall is the tree?
31°
25 ft
9.2
Angles and Radian Measure (pp. 469−476)
Convert the degree measure to radians or the radian measure to degrees.
7π
b. —
12
a. 110°
π radians
110° = 110 degrees —
180 degrees
(
)
11π
=—
18
7π
12
7π
12
180°
( π radians
)
— = — radians —
= 105°
3. Find one positive angle and one negative angle that are coterminal with 382°.
Convert the degree measure to radians or the radian measure to degrees.
4. 30°
5. 225°
3π
4
6. —
5π
3
7. —
8. A sprinkler system on a farm rotates 140° and sprays water up to 35 meters. Draw a diagram that
shows the region that can be irrigated with the sprinkler. Then find the area of the region.
526
Chapter 9
hsnb_alg2_pe_09ec.indd 526
Trigonometric Ratios and Functions
2/5/15 1:44 PM
9.3
Trigonometric Functions of Any Angle (pp. 477−484)
Evaluate csc 210°.
The reference angle is θ′ = 210° − 180° = 30°. The cosecant function is negative in
Quadrant III, so csc 210° = −csc 30° = −2.
Evaluate the six trigonometric functions of θ.
9.
y
10.
(0, 1)
11.
y
y
(−4, 6)
θ
θ
θ
x
x
x
(24, −7)
Evaluate the function without using a calculator.
12. tan 330°
9.4
13π
6
13. sec(−405°)
11π
3
14. sin —
15. sec —
Graphing Sine and Cosine Functions (pp. 485−494)
1
Identify the amplitude and period of g(x) = — sin 2x. Then graph the function and describe the
2
graph of g as a transformation of the graph of f (x) = sin x.
1
The function is of the form g(x) = a sin bx, where a = — and b = 2. So, the amplitude is
2
1
2π 2π
a = — and the period is — = — = π.
y
2
b
2
π
1
Intercepts: (0, 0); — π, 0 = —, 0 ; (π, 0)
2
2
(
) ( )
π 1
1
1
Maximum: ( ⋅ π, ) = ( , )
4
2
4 2
3
1
3π 1
Minimum: ( ⋅ π, − ) = ( , − )
4
2
4
2
—
—
⋅
—
— —
—
—
0.5
π
4
3π
4
x
−0.5
—
The graph of g is a vertical shrink by a factor of —12 and a horizontal shrink by a factor of —12
of the graph of f.
Identify the amplitude and period of the function. Then graph the function and describe the
graph of g as a transformation of the graph of the parent function.
1
16. g(x) = 8 cos x
17. g(x) = 6 sin πx
18. g(x) = — cos 4x
4
Graph the function.
19. g(x) = cos(x + π) + 2
20. g(x) = −sin x − 4
(
Chapter 9
hsnb_alg2_pe_09ec.indd 527
π
2
21. g(x) = 2 sin x + —
)
Chapter Review
527
2/5/15 1:44 PM
9.5
Graphing Other Trigonometric Functions (pp. 497−504)
a. Graph one period of g(x) = 7 cot πx. Describe the graph of g as a transformation of
the graph of f (x) = cot x.
π
π
The function is of the form g(x) = a cot bx, where a = 7 and b = π. So, the period is — = — = 1.
∣b∣ π
π
π
1
Intercepts: —, 0 = —, 0 = —, 0
2b
2π
2
(
) ( )
) (
y
7
π
π
Asymptotes: x = 0; x = — = —, or x = 1
∣b∣ π
π
π
1
Halfway points: —, a = —, 7 = —, 7 ;
4b
4π
4
(
) (
) ( )
1
2
−7
1 x
( 34bπ, −a ) = ( 43ππ, −7 ) = ( 34, −7 )
—
—
—
The graph of g is a vertical stretch by a factor of 7 and a horizontal shrink by a
1
factor of — of the graph of f.
π
b. Graph one period of g(x) = 9 sec x. Describe the graph of g as a transformation of
the graph of f (x) = sec x.
Step 1 Graph the function y = 9 cos x.
2π
The period is — = 2π.
1
y
18
Step 2 Graph asymptotes of g. Because the
asymptotes of g occur when 9 cos x = 0,
π
π
3π
graph x = −—, x = —, and x = —.
2
2
2
π
2
π
−
2
x
−18
Step 3 Plot the points on g, such as (0, 9) and
(π, −9). Then use the asymptotes to
sketch the curve.
The graph of g is a vertical stretch by a factor of 9 of the graph of f.
Graph one period of the function. Describe the graph of g as a transformation of the graph
of its parent function.
1
2
22. g(x) = tan —x
24. g(x) = 4 tan 3πx
23. g(x) = 2 cot x
Graph the function.
528
26. g(x) = sec —x
27. g(x) = 5 sec πx
28. g(x) = — csc —x
Chapter 9
hsnb_alg2_pe_09ec.indd 528
1
2
25. g(x) = 5 csc x
1
2
π
4
Trigonometric Ratios and Functions
2/5/15 1:44 PM
9.6
Modeling with Trigonometric Functions (pp. 505−512)
Write a function for the sinusoid shown.
Step 1
Step 2
(π2 , 3(
y
4
Find the maximum and minimum values. From
the graph, the maximum value is 3 and the minimum
value is −1.
Identify the vertical shift, k. The value of k is the
mean of the maximum and minimum values.
π
2
7π
6
(π6 , −1(
−2
x
(maximum value) + (minimum value) 3 + (−1) 2
k = ———— = — = — = 1
2
2
2
Step 3
Decide whether the graph should be modeled by a sine or cosine function. Because
the graph crosses the midline y = 1 on the y-axis and then decreases to its minimum value,
the graph is a sine curve with a reflection in the x-axis and no horizontal shift. So, h = 0.
Step 4
Find the amplitude and period.
2π 2π
The period is — = —. So, b = 3.
3
b
The amplitude is
(maximum value) − (minimum value) 3 − (−1) 4
∣ a ∣ = ————
= — = — = 2.
2
2
2
Because the graph is a reflection in the x-axis, a < 0. So, a = −2.
The function is y = −2 sin 3x + 1.
Write a function for the sinusoid.
29.
y
−π
30.
(3π , 1)
1
π
y
−3
(0, −1)
1
3
x
x
3π
(π , −1)
−4
(1, −3)
31. You put a reflector on a spoke of your bicycle wheel. The highest point of the reflector
is 25 inches above the ground, and the lowest point is 2 inches. The reflector makes
1 revolution per second. Write a model for the height h (in inches) of a reflector as a
function of time t (in seconds) given that the reflector is at its lowest point when t = 0.
32. The table shows the monthly precipitation P (in inches) for Bismarck, North Dakota,
where t = 1 represents January. Write a model that gives P as a function of t and
interpret the period of its graph.
t
1
2
3
4
5
6
7
8
9
10
11
12
P
0.5
0.5
0.9
1.5
2.2
2.6
2.6
2.2
1.6
1.3
0.7
0.4
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9.7
Using Trigonometric Identities (pp. 513−518)
cot2 θ
Verify the identity — = csc θ − sin θ.
csc θ
cot2 θ
csc θ
csc2 θ − 1
csc θ
—=—
Pythagorean identity
csc2 θ
1
=—−—
csc θ
csc θ
Write as separate fractions.
1
= csc θ − —
csc θ
Simplify.
= csc θ − sin θ
Reciprocal identity
Simplify the expression.
(sec x + 1)(sec x − 1)
tan x
33. cot2 x − cot2 x cos2 x
34. ——
( π2 )
35. sin — − x tan x
Verify the identity.
( π2 )
cos x sec x
1 + tan x
37. tan — − x cot x = csc2 x − 1
36. —
= cos2 x
2
9.8
Using Sum and Difference Formulas (pp. 519−524)
Find the exact value of sin 105°.
sin 105° = sin(45° + 60°)
Substitute 45° + 60° for 105°.
= sin 45° cos 60° + cos 45° sin 60°
—
—
√2 1 √2 √3
=— —+— —
2 2
2
2
—
⋅
⋅
Sum formula for sine
—
Evaluate.
—
√2 + √6
=—
4
Simplify.
—
—
√2 + √6
The exact value of sin 105° is —.
4
Find the exact value of the expression.
38. sin 75°
π
12
39. tan(−15°)
1
4
40. cos —
3π
2
π
2
3
7
41. Find tan(a + b), given that tan a = — with π < a < — and tan b = — with 0 < b < —.
Solve the equation for 0 ≤ x < 2π.
(
3π
4
)
(
3π
4
)
42. cos x + — + cos x − — = 1
530
Chapter 9
hsnb_alg2_pe_09ec.indd 530
(
π
2
)
43. tan(x + π) + cos x + — = 0
Trigonometric Ratios and Functions
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9
Chapter Test
Verify the identity.
cos2 x + sin2 x
1 + tan x
1 + sin x
cos x
3π
2
(
cos x
1 + sin x
2. — + — = 2 sec x
1. ——
= cos2 x
2
)
3. cos x + — = sin x
4. Evaluate sec(−300°) without using a calculator.
Write a function for the sinusoid.
5.
y
6.
(2, 5)
( 98π , 1(
y
1
π
4
3
π
2
π
5π
4
x
−3
1
2
(1, −1)
(38π , −5(
−5
x
Graph the function. Then describe the graph of g as a transformation of the graph of its
parent function.
1
7. g(x) = −4 tan 2x
8. g(x) = −2 cos —x + 3
9. g(x) = 3 csc πx
3
Convert the degree measure to radians or the radian measure to degrees. Then find one
positive angle and one negative angle that are coterminal with the given angle.
4π
5
10. −50°
8π
3
11. —
12. —
13. Find the arc length and area of a sector with radius r = 13 inches and central angle θ = 40°.
Evaluate the six trigonometric functions of the angle θ.
14.
15.
y
y
θ
θ
x
x
(−1, 0)
(2, −9)
16. In which quadrant does the terminal side of θ lie when cos θ < 0 and tan θ > 0? Explain.
200 ft
17. How tall is the building? Justify your answer.
h
60°
18. The table shows the average daily high temperatures T (in degrees Fahrenheit) in
5 ft
Baltimore, Maryland, where m = 1 represents January. Write a model that gives T as
a function of m and interpret the period of its graph.
m
1
2
3
4
5
6
7
8
9
10
11
12
T
41
45
54
65
74
83
87
85
78
67
56
45
Chapter 9
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Chapter Test
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9
Cumulative Assessment
1. Which expressions are equivalent to 1?
tan x sec x cos x
π
cos — − x csc x
2
cos2(−x) tan2 x
sin (−x)
sin2 x + cos2 x
(
——
2
)
2. Which rational expression represents the ratio of the perimeter to the area of the
playground shown in the diagram?
9
A —
○
7x
11
B —
○
14x
2x yd
x yd
2x yd
1
C —
○
x
6x yd
1
D —
○
2x
3. The chart shows the average monthly temperatures (in degrees Fahrenheit) and the
gas usages (in cubic feet) of a household for 12 months.
a. Use a graphing calculator to find
trigonometric models for the average
temperature y1 as a function of time and
the gas usage y2 (in thousands of cubic
feet) as a function of time. Let t = 1
represent January.
January
February
March
April
32°F
21°F
15°F
22°F
20,000 ft3
27,000 ft3
23,000 ft3
22,000 ft3
May
June
July
August
35°F
49°F
62°F
78°F
21,000 ft3
14,000 ft3
8,000 ft3
9,000 ft3
September
October
November
December
71°F
63°F
55°F
40°F
13,000 ft3
15,000 ft3
19,000 ft3
23,000 ft3
b. Graph the two regression equations
in the same coordinate plane on your
graphing calculator. Describe the
relationship between the graphs.
4. Evaluate each logarithm using log2 5 ≈ 2.322 and log2 3 ≈ 1.585, if necessary.
Then order the logarithms by value from least to greatest.
a. log 1000
b. log2 15
c. ln e
d. log2 9
e.
532
log2 —53
Chapter 9
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f. log2 1
Trigonometric Ratios and Functions
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5. Which function is not represented by the graph?
A y = 5 sin x
○
π
B y = 5 cos — − x
○
2
(
π
C y = 5 cos x + —
○
2
(
y
)
g
5
)
x
3π
−
2
π
2
D y = −5 sin(x + π)
○
6. Complete each statement with < or > so that each statement is true.
a. θ
b. tan θ
c. θ′
y
3 radians
s = 4π
0
θ
r=6
x
45°
7. Use the Rational Root Theorem and the graph to find all the real zeros of the function
f (x) = 2x3 − x2 − 13x − 6.
y
f
5
2
x
−10
−20
5π
6
8. Your friend claims −210° is coterminal with the angle —. Is your friend correct?
Explain your reasoning.
9. Company A and Company B offer the same starting annual salary of $20,000.
Company A gives a $1000 raise each year. Company B gives a 4% raise each year.
a. Write rules giving the salaries an and bn for your nth year of employment
at Company A and Company B, respectively. Tell whether the sequence
represented by each rule is arithmetic, geometric, or neither.
b. Graph each sequence in the same coordinate plane.
c. Under what conditions would you choose to work for Company B?
d. After 20 years of employment, compare your total earnings.
Chapter 9
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Cumulative Assessment
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