Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1
Instructor’s Solutions Manual, Section 5.1
Exercise 1
Solutions to Exercises, Section 5.1
1
1. Find all numbers t such that ( 3 , t) is a point on the unit circle.
1
solution For ( 3 , t) to be a point on the unit circle means that the sum
of the squares of the coordinates equals 1. In other words,
1 2
3
+ t 2 = 1.
√
8
8
This simplifies to the equation t 2 = 9 , which implies that t = 3 or
√
√
√
√
√
√
8
t = − 3 . Because 8 = 4 · 2 = 4 · 2 = 2 2, we can rewrite this as
t=
√
2 2
3
√
2
or t = − 2 3 .
Student Solution Manual, Section 5.1
Exercise 3
3. Find all numbers t such that (t, − 25 ) is a point on the unit circle.
2
solution For (t, − 5 ) to be a point on the unit circle means that the
sum of the squares of the coordinates equals 1. In other words,
2 2
t 2 + − 5 = 1.
This simplifies to the equation t 2 =
t=
√
21
− 5 .
21
25 ,
which implies that t =
√
21
5
or
Instructor’s Solutions Manual, Section 5.1
Exercise 5
5. Find the points where the line through the origin with slope 3 intersects
the unit circle.
solution The line through the origin with slope 3 is characterized by
the equation y = 3x. Substituting this value for y into the equation for
the unit circle (x 2 + y 2 = 1) gives
x 2 + (3x)2 = 1,
√
10
√
10
which simplifies to the equation 10x 2 = 1. Thus x = 10 or x = − 10 .
Using each
of these values√of x along
with the equation y = 3x gives the
√
√10 3√10 10
3 10 points 10 , 10 and − 10 , − 10 as the points of intersection of the
line y = 3x and the unit circle.
Student Solution Manual, Section 5.1
Exercise 7
7. Suppose an ant walks counterclockwise on the unit circle from the point
(1, 0) to the endpoint of the radius that forms an angle of 70◦ with the
positive horizontal axis. How far has the ant walked?
solution We need to find the length of the circular arc on the unit circle
70π
7π
corresponding to a 70◦ angle. This length equals 180 , which equals 18 .
Instructor’s Solutions Manual, Section 5.1
Exercise 9
9. What angle corresponds to a circular arc on the unit circle with length
π
5?
solution Let θ be such that the angle of θ degrees corresponds to an
π
θπ
π
arc on the unit circle with length 5 . Thus 180 = 5 . Solving this equation
for θ, we get θ = 36. Thus the angle in question is 36◦ .
Student Solution Manual, Section 5.1
11.
Exercise 11
What angle corresponds to a circular arc on the unit circle with length
5
2?
solution Let θ be such that the angle of θ degrees corresponds to an arc
θπ
on the unit circle with length 52 . Thus 180
= 52 . Solving this equation for θ,
we get θ = 450
π . Thus the angle in question is
equal to 143.2◦ .
450 ◦
π ,
which is approximately
Instructor’s Solutions Manual, Section 5.1
Exercise 13
13. Find the lengths of√ both
circular arcs on the unit circle connecting the
2 √2 points (1, 0) and 2 , 2 .
√ √ solution The radius of the unit circle ending at the point 22 , 22
horizontal axis. One of the cirmakes an angle of 45◦ with the positive
√2 √2 cular arcs connecting (1, 0) and 2 , 2 is shown below as the thickened
√2 √2 circular arc; the other circular arc connecting (1, 0) and 2 , 2 is the
unthickened part of the unit circle below.
45π
π
The length of the thickened arc below is 180 , which equals 4 . The entire
unit circle has length 2π . Thus the length of the other circular arc below
π
7π
is 2π − 4 , which equals 4 .
45
1
The thickened circular arc has length
7π
The other circular arc has length 4 .
π
4.
Student Solution Manual, Section 5.1
Exercise 15
For each of the angles in Exercises 15–20, find the endpoint of the radius
of the unit circle that makes the given angle with the positive horizontal
axis.
15. 120◦
solution The radius making a 120◦ angle with the positive horizontal
axis is shown below. The angle from this radius to the negative horizontal axis equals 180◦ − 120◦ , which equals 60◦ as shown in the figure
below. Drop a perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. We
already know that one angle of this right triangle is 60◦ ; thus the other
angle must be 30◦ , as labeled below:
120
30
60
1
The side of the right triangle opposite the 30◦ angle has length
√
3
2 .
1
2;
the
Looking
side of the right triangle opposite the 60◦ angle has length
at the figure above, we see that the first coordinate of the endpoint of the
Instructor’s Solutions Manual, Section 5.1
Exercise 15
radius is the negative of the length of the side opposite the 30◦ angle,
and the second coordinate of the endpoint of the radius is the length
of the√ side opposite the 60◦ angle. Thus the endpoint of the radius is
1 3
−2, 2 .
Student Solution Manual, Section 5.1
Exercise 17
17. −30◦
solution The radius making a −30◦ angle with the positive horizontal axis is shown below. Draw a perpendicular line segment from the
endpoint of the radius to the horizontal axis, forming a right triangle as
shown below. We already know that one angle of this right triangle is
30◦ ; thus the other angle must be 60◦ , as labeled below.
The side of the right triangle opposite the 30◦ angle has length
√
3
2 .
◦
1
2;
the
Looking
side of the right triangle opposite the 60 angle has length
at the figure below, we see that the first coordinate of the endpoint of the
radius is the length of the side opposite the 60◦ angle, and the second
coordinate of the endpoint of the radius is the negative of the length
of the side opposite the 30◦ angle. Thus the endpoint of the radius is
√3
1
2 , −2 .
30
1
60
Instructor’s Solutions Manual, Section 5.1
Exercise 19
19. 390◦
solution The radius making a 390◦ angle with the positive horizontal
axis is obtained by starting at the horizontal axis, making one complete
counterclockwise rotation, and then continuing for another 30◦ . The
resulting radius is shown below. Drop a perpendicular line segment from
the endpoint of the radius to the horizontal axis, forming a right triangle
as shown below. We already know that one angle of this right triangle is
30◦ ; thus the other angle must be 60◦ , as labeled below.
The side of the right triangle opposite the 30◦ angle has length 12 ; the side
√
3
opposite the 60◦ angle has length 2 . Looking at the figure below, we see
that the first coordinate of the endpoint of the radius is the length of the
side opposite the 60◦ angle, and the second coordinate of the endpoint
of the radius is the length √of the side opposite the 30◦ angle. Thus the
3 1
endpoint of the radius is 2 , 2 .
60
30
1
Student Solution Manual, Section 5.1
Exercise 21
For Exercises 21–26, find the angle the radius of the unit circle ending
at the given point makes with the positive horizontal axis. Among the
infinitely many possible correct solutions, choose the one with the smallest
absolute value.
1 √3 21. − 2 , 2
√ solution Draw the radius whose endpoint is − 12 , 23 . Drop a perpendicular line segment from the endpoint of the radius to the horizontal
axis, forming a right triangle. The hypotenuse of this right triangle is a
radius of the unit circle and thus has length 1. The horizontal
side has
√
1
3
length 2 and the vertical side of this triangle has length 2 because the
1 √3 endpoint of the radius is − 2 , 2 .
120
30
60
1
Thus we have a 30◦ - 60◦ - 90◦ triangle, with the 30◦ angle opposite the
1
horizontal side of length 2 , as labeled above. Because 180 − 60 = 120,
Instructor’s Solutions Manual, Section 5.1
Exercise 21
the radius makes a 120◦ angle with the positive horizontal axis, as shown
above.
In addition to making a 120◦ angle with the positive horizontal axis, this
radius also makes with the positive horizontal axis angles of 480◦ , 840◦ ,
and so on. This radius also makes with the positive horizontal axis angles
of −240◦ , −600◦ , and so on. But of all the possible choices for this angle,
the one with the smallest absolute value is 120◦ .
Student Solution Manual, Section 5.1
23.
√2
2
,−
Exercise 23
√
2
2
√ √
solution Draw the radius whose endpoint is 22 , − 22 . Draw a perpendicular line segment from the endpoint of the radius to the horizontal
axis, forming a right triangle. The hypotenuse of this right triangle is a
radius of
√ the unit circle and thus has length 1. The horizontal
√ side has
2
2
length 2 and the vertical side of this triangle also has length 2 because
√ √2
2
the endpoint of the radius is 2 , − 2 .
45
1
45
Thus we have here an isosceles right triangle, with two angles of 45◦ as
labeled above.
In addition to making a −45◦ angle with the positive horizontal axis,
this radius also makes with the positive horizontal axis angles of 315◦ ,
675◦ , and so on. This radius also makes with the positive horizontal axis
angles of −405◦ , −765◦ , and so on. But of all the possible choices for
this angle, the one with the smallest absolute value is −45◦ .
Instructor’s Solutions Manual, Section 5.1
25.
−
Exercise 25
√
√ 2
2
2 ,− 2
√ √
solution Draw the radius whose endpoint is − 22 , − 22 . Draw a perpendicular line segment from the endpoint of the radius to the horizontal
axis, forming a right triangle. The hypotenuse of this right triangle is a
radius of
√ the unit circle and thus has length 1. The horizontal
√ side has
2
2
length 2 and the vertical side of this triangle also has length 2 because
√ √2
2
the endpoint of the radius is − 2 , − 2 .
45
45
1
135
Thus we have here an isosceles right triangle, with two angles of 45◦ as
labeled above. Because the radius makes a 45◦ angle with the negative
horizontal axis, it makes a −135◦ angle with the positive horizontal axis,
as shown below (because 135◦ = 180◦ − 45◦ ).
In addition to making a −135◦ angle with the positive horizontal axis,
this radius also makes with the positive horizontal axis angles of 225◦ ,
585◦ , and so on. This radius also makes with the positive horizontal axis
Student Solution Manual, Section 5.1
Exercise 25
angles of −495◦ , −855◦ , and so on. But of all the possible choices for
this angle, the one with the smallest absolute value is −135◦ .
Instructor’s Solutions Manual, Section 5.1
Exercise 27
27. Find the lengths
of both circular arcs on the unit circle connecting the
1 √3 point 2 , 2 and the point that makes an angle of 130◦ with the positive
horizontal axis.
1 √3 solution The radius of the unit circle ending at the point 2 , 2 makes
with the positive horizontal axis. One of the circular arcs
an angle of 60◦ √
1 3
connecting 2 , 2 and the point that makes an angle of 130◦ with the
positive horizontal axis is shown below as the thickened circular arc; the
other circular arc connecting these two points is the unthickened part of
the unit circle below.
The thickened arc below corresponds to an angle of 70◦ (because 70◦ =
70π
130◦ − 60◦ ). Thus the length of the thickened arc below is 180 , which
7π
equals 18 . The entire unit circle has length 2π . Thus the length of the
7π
29π
other circular arc below is 2π − 18 , which equals 18 .
60
130
7π
1
The thickened circular arc has length 18 .
29π
The other circular arc has length 18 .
Student Solution Manual, Section 5.1
Exercise 29
29. Find the √
lengths
√ of both circular arcs on the unit circle connecting the
2
2
point − 2 , − 2 and the point that makes an angle of 125◦ with the
positive horizontal axis.
√ √2
2
solution The radius of the unit circle ending at − 2 , − 2 makes an
axis
225◦ = 180◦ +45◦ ).
angle of 225◦ with the positive horizontal
√ (because
√2
2
One of the circular arcs connecting − 2 , − 2 and the point that makes
an angle of 125◦ with the positive horizontal axis is shown below as the
thickened circular arc; the other circular arc connecting these two points
is the unthickened part of the unit circle below.
The thickened arc below corresponds to an angle of 100◦ (because 100◦ =
100π
225◦ − 125◦ ). Thus the length of the thickened arc below is 180 , which
5π
equals 9 . The entire unit circle has length 2π . Thus the length of the
5π
13π
other circular arc below is 2π − 9 , which equals 9 .
125
225
1
The thickened circular arc has length
13π
The other circular arc has length 9 .
5π
9 .
Instructor’s Solutions Manual, Section 5.1
Exercise 31
31. What is the slope of the radius of the unit circle that has a 30◦ angle with
the positive horizontal axis?
solution The radius of the unit circle that has a 30◦ angle with the
positive
horizontal axis has its initial point at (0, 0) and its endpoint at
√3 1 ,
.
Thus
the slope of this radius is
2 2
1
−0
√2
3
2 −0
which equals
√1 ,
3
which equals
√
3
3 .
,
Student Solution Manual, Section 5.2
Exercise 1
Solutions to Exercises, Section 5.2
In Exercises 1–8, convert each angle to radians.
1. 15◦
solution Start with the equation
360◦ = 2π radians.
Divide both sides by 360 to obtain
1◦ =
π
radians.
180
Now multiply both sides by 15, obtaining
15◦ =
π
15π
radians =
radians.
180
12
Instructor’s Solutions Manual, Section 5.2
3. −45◦
solution Start with the equation
360◦ = 2π radians.
Divide both sides by 360 to obtain
1◦ =
π
radians.
180
Now multiply both sides by −45, obtaining
−45◦ = −
π
45π
radians = − radians.
4
180
Exercise 3
Student Solution Manual, Section 5.2
Exercise 5
5. 270◦
solution Start with the equation
360◦ = 2π radians.
Divide both sides by 360 to obtain
1◦ =
π
radians.
180
Now multiply both sides by 270, obtaining
270◦ =
3π
270π
radians.
radians =
2
180
Instructor’s Solutions Manual, Section 5.2
7. 1080◦
solution Start with the equation
360◦ = 2π radians.
Divide both sides by 360 to obtain
1◦ =
π
radians.
180
Now multiply both sides by 1080, obtaining
1080◦ =
1080π
radians = 6π radians.
180
Exercise 7
Student Solution Manual, Section 5.2
In Exercises 9–16, convert each angle to degrees.
9. 4π radians
solution Start with the equation
2π radians = 360◦ .
Multiply both sides by 2, obtaining
4π radians = 2 · 360◦ = 720◦ .
Exercise 9
Instructor’s Solutions Manual, Section 5.2
11.
π
9
radians
solution Start with the equation
2π radians = 360◦ .
Divide both sides by 2 to obtain
π radians = 180◦ .
Now divide both sides by 9, obtaining
180 ◦
π
radians =
= 20◦ .
9
9
Exercise 11
Student Solution Manual, Section 5.2
Exercise 13
13. 3 radians
solution Start with the equation
2π radians = 360◦ .
Divide both sides by 2π to obtain
1 radian =
180 ◦
.
π
Now multiply both sides by 3, obtaining
3 radians = 3 ·
540 ◦
180 ◦
=
.
π
π
Instructor’s Solutions Manual, Section 5.2
15. − 2π
3 radians
solution Start with the equation
2π radians = 360◦ .
Divide both sides by 2 to obtain
π radians = 180◦ .
2
Now multiply both sides by − 3 , obtaining
−
2
2π
radians = − · 180◦ = −120◦ .
3
3
Exercise 15
Student Solution Manual, Section 5.2
Exercise 17
17. Suppose an ant walks counterclockwise on the unit circle from the point
5π
(0, 1) to the endpoint of the radius that forms an angle of 4 radians
with the positive horizontal axis. How far has the ant walked?
solution The radius whose endpoint equals (0, 1) makes an angle of
π
2 radians with the positive horizontal axis. This radius corresponds to
the smaller angle shown below.
5π
Because 4 = π + π4 , the radius that forms an angle of 5π
4 radians with
π
the positive horizontal axis lies 4 radians beyond the negative horizontal axis (half-way between the negative horizontal axis and the negative
vertical axis). Thus the ant ends its walk at the endpoint of the radius
corresponding to the larger angle shown below:
1
The ant walks along the thickened circular arc shown above. This circular
5π
π
3π
arc corresponds to an angle of 4 − 2 radians, which equals 4 radians.
3π
Thus the distance walked by the ant is 4 .
Instructor’s Solutions Manual, Section 5.2
19.
Exercise 19
Find the lengths of both circular arcs of the unit circle connecting
the point (1, 0) and the endpoint of the radius that makes an angle of 3
radians with the positive horizontal axis.
solution Because 3 is a bit less than π , the radius that makes an angle
of 3 radians with the positive horizontal axis lies a bit above the negative
horizontal axis, as shown below. The thickened circular arc corresponds
to an angle of 3 radians and thus has length 3. The entire unit circle has
length 2π . Thus the length of the other circular arc is 2π − 3, which is
approximately 3.28.
1
Student Solution Manual, Section 5.2
21.
Exercise 21
Find √the lengths
of both circular arcs of the unit circle connecting the
√ 2
2
point 2 , − 2 and the point whose radius makes an angle of 1 radian
with the positive horizontal axis.
√ √2
2
solution The radius of the unit circle whose endpoint equals 2 , − 2
π
makes an angle of − 4 radians with the positive horizontal axis, as shown
with the clockwise arrow below. The radius that makes an angle of 1
radian with the positive horizontal axis is shown with a counterclockwise
arrow.
1
π
Thus the thickened circular arc above corresponds to an angle of 1 + 4
and thus has length 1 + π4 , which is approximately 1.79. The entire unit
circle has length 2π . Thus the length of the other circular arc below is
π
7π
2π − (1 + 4 ), which equals 4 − 1, which is approximately 4.50.
Instructor’s Solutions Manual, Section 5.2
23. For a 16-inch pizza, find the area of a slice with angle
Exercise 23
3
4
radians.
solution Pizzas are measured by their diameters; thus this pizza has
1
3
a radius of 8 inches. Thus the area of the slice is 2 · 4 · 82 , which equals
24 square inches.
Student Solution Manual, Section 5.2
Exercise 25
25. Suppose a slice of a 12-inch pizza has an area of 20 square inches. What
is the angle of this slice?
solution This pizza has a radius of 6 inches. Let θ denote the angle
of this slice, measured in radians. Then
1
20 = 2 θ · 62 .
Solving this equation for θ, we get θ =
10
9
radians.
Instructor’s Solutions Manual, Section 5.2
27.
Exercise 27
Suppose a slice of pizza with an angle of 56 radians has an area of 21
square inches. What is the diameter of this pizza?
solution Let r denote the radius of this pizza. Thus
21 =
1
2
· 56 r 2 .
252
Solving this equation for r , we get r =
5 ≈ 7.1. Thus the diameter of
the pizza is approximately 14.2 inches.
Student Solution Manual, Section 5.2
Exercise 29
For each of the angles in Exercises 29–34, find the endpoint of the radius
of the unit circle that makes the given angle with the positive horizontal
axis.
29.
5π
6
radians
solution For this exercise it may be easier to convert to degrees. Thus
5π
we translate 6 radians to 150◦ .
The radius making a 150◦ angle with the positive horizontal axis is shown
below. The angle from this radius to the negative horizontal axis equals
180◦ − 150◦ , which equals 30◦ as shown in the figure below. Drop a
perpendicular line segment from the endpoint of the radius to the horizontal axis, forming a right triangle as shown below. We already know
that one angle of this right triangle is 30◦ ; thus the other angle must be
60◦ , as labeled below.
The side of the right triangle opposite the 30◦ angle has length
◦
√
3
2 .
1
2;
the
Looking
side of the right triangle opposite the 60 angle has length
at the figure below, we see that the first coordinate of the endpoint of the
radius is the negative of the length of the side opposite the 60◦ angle,
and the second coordinate of the endpoint of the radius is the length
of the
side opposite the 30◦ angle. Thus the endpoint of the radius is
√3 1 − 2 ,2 .
Instructor’s Solutions Manual, Section 5.2
Exercise 29
150
60
30
1
Student Solution Manual, Section 5.2
Exercise 31
31. − π4 radians
solution For this exercise it may be easier to convert to degrees. Thus
π
we translate − 4 radians to −45◦ .
The radius making a −45◦ angle with the positive horizontal axis is
shown below. Draw a perpendicular line segment from the endpoint
of the radius to the horizontal axis, forming a right triangle as shown
below. We already know that one angle of this right triangle is 45◦ ; thus
the other angle must also be 45◦ , as labeled below.
The hypotenuse of this right triangle is a radius of the unit
circle and
√
2
thus has length 1. The other two sides each have length 2 . Looking at
the figure√below, we see that the first coordinate of the endpoint of the
2
radius is 2 and the second coordinate of the endpoint of the radius is
√
√ √2
2
2
− 2 . Thus the endpoint of the radius is 2 , − 2 .
45
1
45
Instructor’s Solutions Manual, Section 5.2
33.
5π
2
Exercise 33
radians
5π
π
solution Note that 2 = 2π + 2 . Thus the radius making an angle of
5π
2 radians with the positive horizontal axis is obtained by starting at the
horizontal axis, making one complete counterclockwise rotation (which
π
is 2π radians), and then continuing for another 2 radians. The resulting
radius is shown below. Its endpoint is (0, 1).
1
Student Solution Manual, Section 5.3
Exercise 1
Solutions to Exercises, Section 5.3
Give exact values for the quantities in Exercises 1–10. Do not use a calculator for any of these exercises—otherwise you will likely get decimal
approximations for some solutions rather than exact answers. More importantly, good understanding will come from working these exercises by
hand.
1.
(a) cos 3π
(b) sin 3π
solution Because 3π = 2π + π , an angle of 3π radians (as measured
counterclockwise from the positive horizontal axis) consists of a complete revolution around the circle (2π radians) followed by another π
radians (180◦ ), as shown below. The endpoint of the corresponding radius is (−1, 0). Thus cos 3π = −1 and sin 3π = 0.
1
Instructor’s Solutions Manual, Section 5.3
3.
(a) cos 11π
4
Exercise 3
(b) sin 11π
4
solution Because 11π
= 2π + π2 + π4 , an angle of 11π
radians (as
4
4
measured counterclockwise from the positive horizontal axis) consists
of a complete revolution around the circle (2π radians) followed by anπ
π
other 2 radians (90◦ ), followed by another 4 radians (45◦ ), as shown
√2 √2 below. Hence the endpoint of the corresponding radius is − 2 , 2 .
Thus cos
11π
4
=−
√
2
2
and sin
11π
4
=
√
2
2 .
1
Student Solution Manual, Section 5.3
5.
(a) cos 2π
3
Exercise 5
(b) sin 2π
3
π
π
2π
solution Because 2π
3 = 2 + 6 , an angle of 3 radians (as measured
π
counterclockwise from the positive horizontal axis) consists of 2 radians
π
◦
◦
(90 radians) followed by another 6 radians (30 ), as shown below. The
1 √3 2π
1
endpoint of the corresponding radius is − 2 , 2 . Thus cos 3 = − 2 and
sin
2π
3
=
√
3
2 .
1
Instructor’s Solutions Manual, Section 5.3
7.
(a) cos 210◦
Exercise 7
(b) sin 210◦
solution Because 210 = 180 + 30, an angle of 210◦ (as measured counterclockwise from the positive horizontal axis) consists of 180◦ followed
◦
, as shown below. The √endpoint of the corresponding
by another 30
√3
1
3
1
radius is − 2 , − 2 . Thus cos 210◦ = − 2 and sin 210◦ = − 2 .
1
Student Solution Manual, Section 5.3
9.
Exercise 9
(a) cos 360045◦
(b) sin 360045◦
solution Because 360045 = 360 × 1000 + 45, an angle of 360045◦ (as
measured counterclockwise from the positive horizontal axis) consists
of 1000 complete revolutions around the circle√followed
by another 45◦ .
2 √2 The endpoint of the corresponding radius is 2 , 2 . Thus
cos 360045◦ =
√
2
2
and
sin 360045◦ =
√
2
2 .
Instructor’s Solutions Manual, Section 5.3
Exercise 11
11. Find the smallest number θ larger than 4π such that cos θ = 0.
solution Note that
0 = cos
π
2
= cos
3π
2
= cos
5π
2
= ...
and that the only numbers whose cosine equals 0 are of the form (2n+1)π
,
2
where n is an integer. The smallest number of this form larger than 4π is
9π
9π
2 . Thus 2 is the smallest number larger than 4π whose cosine equals
0.
Student Solution Manual, Section 5.3
Exercise 13
13. Find the four smallest positive numbers θ such that cos θ = 0.
solution Think of a radius of the unit circle whose endpoint is (1, 0). If
this radius moves counterclockwise, forming an angle of θ with the positive horizontal axis, the first coordinate of its endpoint first becomes 0
π
3π
when θ equals 2 (which equals 90◦ ), then again when θ equals 2 (which
5π
equals 270◦ ), then again when θ equals 2 (which equals 360◦ + 90◦ , or
7π
450◦ ), then again when θ equals 2 (which equals 360◦ + 270◦ , or 630◦ ),
and so on. Thus the four smallest positive numbers θ such that cos θ = 0
π 3π 5π
7π
are 2 , 2 , 2 , and 2 .
Instructor’s Solutions Manual, Section 5.3
Exercise 15
15. Find the four smallest positive numbers θ such that sin θ = 1.
solution Think of a radius of the unit circle whose endpoint is (1, 0).
If this radius moves counterclockwise, forming an angle of θ with the
positive horizontal axis, then the second coordinate of its endpoint first
π
becomes 1 when θ equals 2 (which equals 90◦ ), then again when θ equals
5π
9π
◦
◦
◦
2 (which equals 360 +90 , or 450 ), then again when θ equals 2 (which
13π
equals 2 × 360◦ + 90◦ , or 810◦ ), then again when θ equals 2 (which
◦
◦
◦
equals 3 × 360 + 90 , or 1170 ), and so on. Thus the four smallest
π 5π 9π
13π
positive numbers θ such that sin θ = 1 are 2 , 2 , 2 , and 2 .
Student Solution Manual, Section 5.3
Exercise 17
17. Find the four smallest positive numbers θ such that cos θ = −1.
solution Think of a radius of the unit circle whose endpoint is (1, 0).
If this radius moves counterclockwise, forming an angle of θ with the
positive horizontal axis, the first coordinate of its endpoint first becomes
−1 when θ equals π (which equals 180◦ ), then again when θ equals 3π
(which equals 360◦ + 180◦ , or 540◦ ), then again when θ equals 5π (which
equals 2 × 360◦ + 180◦ , or 900◦ ), then again when θ equals 7π (which
equals 3 × 360◦ + 180◦ , or 1260◦ ), and so on. Thus the four smallest
positive numbers θ such that cos θ = −1 are π , 3π , 5π , and 7π .
Instructor’s Solutions Manual, Section 5.3
Exercise 19
19. Find the four smallest positive numbers θ such that sin θ = 12 .
solution Think of a radius of the unit circle whose endpoint is (1, 0). If
this radius moves counterclockwise, forming an angle of θ with the posi1
tive horizontal axis, the second coordinate of its endpoint first becomes 2
π
5π
when θ equals 6 (which equals 30◦ ), then again when θ equals 6 (which
13π
equals 150◦ ), then again when θ equals 6 (which equals 360◦ + 30◦ , or
17π
390◦ ), then again when θ equals 6 (which equals 360◦ + 150◦ , or 510◦ ),
1
and so on. Thus the four smallest positive numbers θ such that sin θ = 2
π 5π 13π
17π
are 6 , 6 , 6 , and 6 .
Student Solution Manual, Section 5.3
21. Suppose 0 < θ <
π
2
Exercise 21
and cos θ = 25 . Evaluate sin θ.
solution We know that
(cos θ)2 + (sin θ)2 = 1.
Thus
(sin θ)2 = 1 − (cos θ)2
=1−
=
2 2
5
21
.
25
Because 0 < θ < π2 , we know that sin θ > 0. Thus taking square roots of
both sides of the equation above gives
√
21
.
sin θ =
5
Instructor’s Solutions Manual, Section 5.3
23. Suppose
π
2
Exercise 23
< θ < π and sin θ = 29 . Evaluate cos θ.
solution We know that
(cos θ)2 + (sin θ)2 = 1.
Thus
(cos θ)2 = 1 − (sin θ)2
=1−
=
2 2
9
77
.
81
Because π2 < θ < π , we know that cos θ < 0. Thus taking square roots
of both sides of the equation above gives
√
77
.
cos θ = −
9
Student Solution Manual, Section 5.3
25.
Exercise 25
Suppose − π2 < θ < 0 and cos θ = 0.1. Evaluate sin θ.
solution We know that
(cos θ)2 + (sin θ)2 = 1.
Thus
(sin θ)2 = 1 − (cos θ)2
= 1 − (0.1)2
= 0.99.
π
Because − 2 < θ < 0, we know that sin θ < 0. Thus taking square roots
of both sides of the equation above gives
√
sin θ = − 0.99 ≈ −0.995.
Instructor’s Solutions Manual, Section 5.3
Exercise 27
27. Find the smallest number x such that sin(ex ) = 0.
solution Note that ex is an increasing function. Because ex is positive
for every real number x, and because π is the smallest positive number
whose sine equals 0, we want to choose x so that ex = π . Thus x = ln π .
Student Solution Manual, Section 5.3
29.
Exercise 29
Find the smallest positive number x such that
sin(x 2 + x + 4) = 0.
solution Note that x 2 + x + 4 is an increasing function on the interval
[0, ∞). If x is positive, then x 2 + x + 4 > 4. Because 4 is larger than π
but less than 2π , the smallest number bigger than 4 whose sine equals
0 is 2π . Thus we want to choose x so that x 2 + x + 4 = 2π . In other
words, we need to solve the equation
x 2 + x + (4 − 2π ) = 0.
Using the quadratic formula, we see that the solutions to this equation
are
√
−1 ± 8π − 15
.
x=
2
A calculator shows that choosing the plus sign in the equation above
gives x ≈ 1.0916 and choosing the minus sign gives x ≈ −2.0916. We
seek only positive values of x, and thus we choose the plus sign in the
equation above, getting x ≈ 1.0916.
Instructor’s Solutions Manual, Section 5.4
Exercise 1
Solutions to Exercises, Section 5.4
1. Find the four smallest positive numbers θ such that tan θ = 1.
solution Think of a radius of the unit circle whose endpoint is (1, 0).
If this radius moves counterclockwise, forming an angle of θ with the
positive horizontal axis, then the first and second coordinates of its endpoint first become equal (which is equivalent to having tan θ = 1) when θ
π
5π
equals 4 (which equals 45◦ ), then again when θ equals 4 (which equals
9π
◦
◦
225 ), then again when θ equals 4 (which equals 360 + 45◦ , or 405◦ ),
13π
then again when θ equals 4 (which equals 360◦ + 225◦ , or 585◦ ), and
so on.
Thus the four smallest positive numbers θ such that tan θ = 1 are
9π
13π
4 , and 4 .
π 5π
4, 4 ,
Student Solution Manual, Section 5.4
3. Suppose 0 < θ <
π
2
Exercise 3
and cos θ = 15 . Evaluate:
(a) sin θ
(b) tan θ
solution The figure below gives a sketch of the angle involved in this
exercise:
Θ
1
The angle between 0 and
1
equals 5 .
π
2
whose cosine
(a) We know that
(cos θ)2 + (sin θ)2 = 1.
Thus
1 2
5
+ (sin θ)2 = 1. Solving this equation for (sin θ)2 gives
(sin θ)2 =
24
.
25
The sketch above shows that sin θ > 0. Thus taking square roots of both
sides of the equation above gives
Instructor’s Solutions Manual, Section 5.4
Exercise 3
√
√
√
2 6
24
4·6
=
=
.
sin θ =
5
5
5
(b)
sin θ
=
tan θ =
cos θ
√
2 6
5
1
5
√
= 2 6.
Student Solution Manual, Section 5.4
5. Suppose
π
2
Exercise 5
< θ < π and sin θ = 23 . Evaluate:
(a) cos θ
(b) tan θ
solution The figure below gives a sketch of the angle involved in this
exercise:
Θ
1
The angle between
2
equals 3 .
π
2
and π whose sine
(a) We know that
(cos θ)2 + (sin θ)2 = 1.
Thus (cos θ)2 +
2 2
3
= 1. Solving this equation for (cos θ)2 gives
(cos θ)2 =
5
.
9
The sketch above shows that cos θ < 0. Thus taking square roots of both
sides of the equation above gives
Instructor’s Solutions Manual, Section 5.4
Exercise 5
√
5
.
cos θ = −
3
(b)
√
2
2 5
2
sin θ
3
= − √5 = − √ = −
.
tan θ =
cos θ
5
5
3
Student Solution Manual, Section 5.4
Exercise 7
7. Suppose − π2 < θ < 0 and cos θ = 45 . Evaluate:
(a) sin θ
(b) tan θ
solution The figure below gives a sketch of the angle involved in this
exercise:
Θ
1
The angle between − π2 and 0 whose
4
cosine equals 5 .
(a) We know that
(cos θ)2 + (sin θ)2 = 1.
Thus
4 2
5
+ (sin θ)2 = 1. Solving this equation for (sin θ)2 gives
(sin θ)2 =
9
.
25
The sketch above shows that sin θ < 0. Thus taking square roots of both
sides of the equation above gives
Instructor’s Solutions Manual, Section 5.4
Exercise 7
3
sin θ = − .
5
(b)
3
tan θ =
3
sin θ
= − 54 = − .
cos θ
4
5
Student Solution Manual, Section 5.4
9. Suppose 0 < θ <
π
2
Exercise 9
and tan θ = 14 . Evaluate:
(a) cos θ
(b) sin θ
solution The figure below gives a sketch of the angle involved in this
exercise:
Θ
1
The angle between 0 and
1
tangent equals 4 .
(a) Rewrite the equation tan θ = 14 in the form
sides of this equation by cos θ, we get
sin θ =
1
4
sin θ
cos θ
=
1
4.
π
2
whose
Multiplying both
cos θ.
Substitute this expression for sin θ into the equation (cos θ)2 +(sin θ)2 =
1, getting
1
(cos θ)2 + 16 (cos θ)2 = 1,
which is equivalent to
Instructor’s Solutions Manual, Section 5.4
Exercise 9
(cos θ)2 =
16
.
17
The sketch above shows that cos θ > 0. Thus taking square roots of both
sides of the equation above gives
√
4 17
4
=
.
cos θ = √
17
17
(b) We have already noted that sin θ =
1
4
cos θ. Thus
√
17
.
sin θ =
17
Student Solution Manual, Section 5.4
Exercise 11
11. Suppose − π2 < θ < 0 and tan θ = −3. Evaluate:
(a) cos θ
(b) sin θ
solution The figure below gives a sketch of the angle involved in this
exercise:
Θ
1
The angle between − π2 and 0 whose
tangent equals −3.
(a) Rewrite the equation tan θ = −3 in the form
sides of this equation by cos θ, we get
sin θ
cos θ
= −3. Multiplying both
sin θ = −3 cos θ.
Substitute this expression for sin θ into the equation (cos θ)2 +(sin θ)2 =
1, getting
(cos θ)2 + 9(cos θ)2 = 1,
which is equivalent to
Instructor’s Solutions Manual, Section 5.4
(cos θ)2 =
Exercise 11
1
.
10
The sketch above shows that cos θ > 0. Thus taking square roots of both
sides of the equation above gives
√
10
1
=
.
cos θ = √
10
10
(b) We have already noted that sin θ = −3 cos θ. Thus
√
3 10
.
sin θ = −
10
Student Solution Manual, Section 5.4
Given that
cos 15◦ =
√
2+ 3
2
Exercise 13
and
sin 22.5◦ =
√
2− 2
2
,
in Exercises 13–22 find exact expressions for the indicated quantities.
[These values for cos 15◦ and sin 22.5◦ will be derived in Examples 4 and
5 in Section 6.3.]
13. sin 15◦
solution We know that
(cos 15◦ )2 + (sin 15◦ )2 = 1.
Thus
(sin 15◦ )2 = 1 − (cos 15◦ )2
2 + √3 2
=1−
2
√
2+ 3
=1−
4
√
2− 3
.
=
4
Because sin 15◦ > 0, taking square roots of both sides of the equation
above gives
√
2− 3
◦
.
sin 15 =
2
Instructor’s Solutions Manual, Section 5.4
15. tan 15◦
solution
sin 15◦
cos 15◦
√
2− 3
= √
2+ 3
√
√
2− 3
2− 3
·
= √
√
2+ 3
2− 3
√
2− 3
√
=
4−3
√
=2− 3
tan 15◦ =
Exercise 15
Student Solution Manual, Section 5.4
Exercise 17
17. cot 15◦
solution
cot 15◦ =
1
tan 15◦
=
1
√
2− 3
√
2+ 3
1
√ ·
√
=
2− 3 2+ 3
√
2+ 3
=
4−3
√
=2+ 3
Instructor’s Solutions Manual, Section 5.4
Exercise 19
19. csc 15◦
solution
csc 15◦ =
1
sin 15◦
= 2
2−
√
3
√
2+ 3
√ ·
√
2− 3
2+ 3
√
2 2+ 3
= √
4−3
√
=2 2+ 3
= 2
Student Solution Manual, Section 5.4
Exercise 21
21. sec 15◦
solution
sec 15◦ =
1
cos 15◦
2
= √
2+ 3
√
2− 3
2
=
√ ·
√
2+ 3
2− 3
√
2 2− 3
= √
4−3
√
=2 2− 3
Instructor’s Solutions Manual, Section 5.4
Exercise 23
π
2 ),
Suppose u and ν are in the interval (0,
tan u = 2
and
with
tan ν = 3.
In Exercises 23–32, find exact expressions for the indicated quantities.
23. cot u
solution
cot u =
=
1
tan u
1
2
Student Solution Manual, Section 5.4
Exercise 25
25. cos u
solution We know that
2 = tan u
=
sin u
.
cos u
To find cos u, make the substitution sin u = 1 − (cos u)2 in the equaπ
tion above (this substitution is valid because we know that 0 < u < 2
and thus sin u > 0), getting
1 − (cos u)2
.
2=
cos u
Now square both sides of the equation above, then multiply both sides
by (cos u)2 and rearrange to get the equation
5(cos u)2 = 1.
π
Because 0 < u < 2 , we see that cos u > 0. Thus taking square roots of
1
both sides of the equation above gives cos u = √5 , which can be rewritten
as cos u =
√
5
5 .
Instructor’s Solutions Manual, Section 5.4
27. sin u
solution
sin u = 1 − (cos u)2
1
= 1−
5
4
=
5
2
= √
5
√
2 5
=
5
Exercise 27
Student Solution Manual, Section 5.4
Exercise 29
29. csc u
solution
1
sin u
√
5
=
2
csc u =
Instructor’s Solutions Manual, Section 5.4
Exercise 31
31. sec u
solution
1
cos u
√
= 5
sec u =
Student Solution Manual, Section 5.4
33.
Exercise 33
Find the smallest number x such that tan ex = 0.
solution Note that ex is an increasing function. Because ex is positive
for every real number x, and because π is the smallest positive number
whose tangent equals 0, we want to choose x so that ex = π . Thus
x = ln π ≈ 1.14473.
Instructor’s Solutions Manual, Section 5.5
Exercise 1
Solutions to Exercises, Section 5.5
Use the right triangle below for Exercises 1–76. This triangle is not drawn
to scale corresponding to the data in the exercises.
c
Ν
b
u
a
1. Suppose a = 2 and b = 7. Evaluate c.
solution The Pythagorean Theorem implies that c 2 = 22 + 72 . Thus
√
c = 22 + 72 = 53.
Student Solution Manual, Section 5.5
3. Suppose a = 2 and b = 7. Evaluate cos u.
solution
√
2 53
a
2
adjacent side
√
=
=
=
cos u =
53
hypotenuse
c
53
Exercise 3
Instructor’s Solutions Manual, Section 5.5
5. Suppose a = 2 and b = 7. Evaluate sin u.
solution
√
7 53
b
7
opposite side
√
=
= =
sin u =
53
hypotenuse
c
53
Exercise 5
Student Solution Manual, Section 5.5
7. Suppose a = 2 and b = 7. Evaluate tan u.
solution
tan u =
b
7
opposite side
=
=
adjacent side
a
2
Exercise 7
Instructor’s Solutions Manual, Section 5.5
9. Suppose a = 2 and b = 7. Evaluate cos ν.
solution
√
7 53
b
7
adjacent side
√
=
= =
cos ν =
53
hypotenuse
c
53
Exercise 9
Student Solution Manual, Section 5.5
11. Suppose a = 2 and b = 7. Evaluate sin ν.
solution
√
2 53
a
2
opposite side
√
=
=
=
sin ν =
53
hypotenuse
c
53
Exercise 11
Instructor’s Solutions Manual, Section 5.5
13. Suppose a = 2 and b = 7. Evaluate tan ν.
solution
tan ν =
a
2
opposite side
=
=
adjacent side
b
7
Exercise 13
Student Solution Manual, Section 5.5
Exercise 15
15. Suppose b = 2 and c = 7. Evaluate a.
solution The Pythagorean Theorem implies that a2 + 22 = 72 . Thus
√
√
√ √
√
a = 72 − 22 = 45 = 9 · 5 = 9 · 5 = 3 5.
Instructor’s Solutions Manual, Section 5.5
17. Suppose b = 2 and c = 7. Evaluate cos u.
solution
√
a
3 5
adjacent side
=
=
cos u =
hypotenuse
c
7
Exercise 17
Student Solution Manual, Section 5.5
19. Suppose b = 2 and c = 7. Evaluate sin u.
solution
sin u =
b
2
opposite side
= =
hypotenuse
c
7
Exercise 19
Instructor’s Solutions Manual, Section 5.5
21. Suppose b = 2 and c = 7. Evaluate tan u.
solution
√
2 5
b
2
opposite side
√
=
=
=
tan u =
15
adjacent side
a
3 5
Exercise 21
Student Solution Manual, Section 5.5
23. Suppose b = 2 and c = 7. Evaluate cos ν.
solution
cos ν =
b
2
adjacent side
= =
hypotenuse
c
7
Exercise 23
Instructor’s Solutions Manual, Section 5.5
25. Suppose b = 2 and c = 7. Evaluate sin ν.
solution
√
a
3 5
opposite side
=
=
sin ν =
hypotenuse
c
7
Exercise 25
Student Solution Manual, Section 5.5
27. Suppose b = 2 and c = 7. Evaluate tan ν.
solution
√
a
3 5
opposite side
=
=
tan ν =
adjacent side
b
2
Exercise 27
Instructor’s Solutions Manual, Section 5.5
29.
Suppose a = 5 and u = 17◦ . Evaluate b.
solution We have
tan 17◦ =
opposite side
b
= .
adjacent side
5
Solving for b, we get b = 5 tan 17◦ ≈ 1.53.
Exercise 29
Student Solution Manual, Section 5.5
31.
Exercise 31
Suppose a = 5 and u = 17◦ . Evaluate c.
solution We have
cos 17◦ =
5
adjacent side
= .
hypotenuse
c
Solving for c, we get
c=
5
≈ 5.23.
cos 17◦
Instructor’s Solutions Manual, Section 5.5
33.
Exercise 33
Suppose u = 17◦ . Evaluate cos ν.
solution Because ν = 90◦ −u, we have ν = 73◦ . Thus cos ν = cos 73◦ ≈
0.292.
Student Solution Manual, Section 5.5
35.
Suppose u = 17◦ . Evaluate sin ν.
solution sin ν = sin 73◦ ≈ 0.956
Exercise 35
Instructor’s Solutions Manual, Section 5.5
37.
Suppose u = 17◦ . Evaluate tan ν.
solution tan ν = tan 73◦ ≈ 3.27
Exercise 37
Student Solution Manual, Section 5.5
39.
Exercise 39
Suppose c = 8 and u = 1 radian. Evaluate a.
solution We have
cos 1 =
a
adjacent side
= .
hypotenuse
8
Solving for a, we get
a = 8 cos 1 ≈ 4.32.
When using a calculator to do the approximation above, be sure that your
calculator is set to operate in radian mode.
Instructor’s Solutions Manual, Section 5.5
41.
Suppose c = 8 and u = 1 radian. Evaluate b.
solution We have
sin 1 =
b
opposite side
= .
hypotenuse
8
Solving for b, we get
b = 8 sin 1 ≈ 6.73.
Exercise 41
Student Solution Manual, Section 5.5
43.
Exercise 43
Suppose u = 1 radian. Evaluate cos ν.
solution Because ν =
π
2
− u, we have ν =
π
π
2
− 1. Thus
cos ν = cos( 2 − 1) ≈ 0.841.
Instructor’s Solutions Manual, Section 5.5
45.
Exercise 45
Suppose u = 1 radian. Evaluate sin ν.
solution
π
sin ν = sin( 2 − 1) ≈ 0.540
Student Solution Manual, Section 5.5
47.
Exercise 47
Suppose u = 1 radian. Evaluate tan ν.
solution
π
tan ν = tan( 2 − 1) ≈ 0.642
Instructor’s Solutions Manual, Section 5.5
Exercise 49
49. Suppose c = 4 and cos u = 15 . Evaluate a.
solution We have
adjacent side
a
1
= cos u =
= .
5
hypotenuse
4
Solving this equation for a, we get
a=
4
.
5
Student Solution Manual, Section 5.5
Exercise 51
51. Suppose c = 4 and cos u = 15 . Evaluate b.
solution The Pythagorean Theorem implies that
b=
16
=4
16 −
25
1
=4
1−
25
4 2
5
+ b2 = 42 . Thus
√
8 6
24
=
.
25
5
Instructor’s Solutions Manual, Section 5.5
Exercise 53
53. Suppose cos u = 15 . Evaluate sin u.
solution
sin u = 1 − (cos u)2 =
1−
=
1
25
√
24
2 6
=
25
5
Student Solution Manual, Section 5.5
Exercise 55
55. Suppose cos u = 15 . Evaluate tan u.
solution
sin u
=
tan u =
cos u
√
2 6
5
1
5
√
=2 6
Instructor’s Solutions Manual, Section 5.5
57. Suppose cos u = 15 . Evaluate cos ν.
solution
√
2 6
b
cos ν = = sin u =
c
5
Exercise 57
Student Solution Manual, Section 5.5
Exercise 59
59. Suppose cos u = 15 . Evaluate sin ν.
solution
sin ν =
1
a
= cos u =
c
5
Instructor’s Solutions Manual, Section 5.5
Exercise 61
61. Suppose cos u = 15 . Evaluate tan ν.
solution
sin ν
=
tan ν =
cos ν
1
5
√
2 6
5
√
6
1
= √ =
12
2 6
Student Solution Manual, Section 5.5
Exercise 63
63. Suppose b = 4 and sin ν = 13 . Evaluate a.
solution We have
opposite side
a
1
= sin ν =
= .
3
hypotenuse
c
Thus
c = 3a.
By the Pythagorean Theorem, we also have
c 2 = a2 + 16.
Substituting 3a for c in this equation gives
9a2 = a2 + 16.
Solving the equation above for a shows that a =
√
2.
Instructor’s Solutions Manual, Section 5.5
65. Suppose b = 4 and sin ν = 13 . Evaluate c.
solution We have
Thus
a
1
= sin ν = .
3
c
√
c = 3a = 3 2.
Exercise 65
Student Solution Manual, Section 5.5
Exercise 67
67. Suppose sin ν = 13 . Evaluate cos u.
solution
cos u =
1
a
= sin ν =
c
3
Instructor’s Solutions Manual, Section 5.5
Exercise 69
69. Suppose sin ν = 13 . Evaluate sin u.
solution
sin u = 1 − (cos u)2 =
1−
=
1 2
3
√
8
2 2
=
9
3
Student Solution Manual, Section 5.5
Exercise 71
71. Suppose sin ν = 13 . Evaluate tan u.
solution
sin u
=
tan u =
cos u
√
2 2
3
1
3
√
=2 2
Instructor’s Solutions Manual, Section 5.5
Exercise 73
73. Suppose sin ν = 13 . Evaluate cos ν.
solution
cos ν = 1 − (sin ν)2 =
1−
=
1 2
3
√
8
2 2
=
9
3
Student Solution Manual, Section 5.5
Exercise 75
75. Suppose sin ν = 13 . Evaluate tan ν.
solution
sin ν
=
tan ν =
cos ν
1
3
√
2 2
3
√
2
1
= √ =
4
2 2
Instructor’s Solutions Manual, Section 5.5
77.
Exercise 77
Suppose a 25-foot ladder is leaning against a wall, making a 63◦ angle
with the ground (as measured from a perpendicular line from the base
of the ladder to the wall). How high up the wall is the end of the ladder?
solution
In the sketch here, the vertical line represents the wall
and the hypotenuse represents the ladder. As labeled
here, the ladder touches the wall at height b; thus we
need to evaluate b.
25
b
63
We have sin 63◦ =
b
25 .
Solving this equation for b, we get
b = 25 sin 63◦ ≈ 22.28.
Thus the ladder touches the wall at a height of approximately 22.28 feet.
Because 0.28 × 12 = 3.36, this is approximately 22 feet, 3 inches.
Student Solution Manual, Section 5.5
79.
Exercise 79
Suppose you need to find the height of a tall building. Standing 20
meters away from the base of the building, you aim a laser pointer at
the closest part of the top of the building. You measure that the laser
pointer is 4◦ tilted from pointing straight up. The laser pointer is held 2
meters above the ground. How tall is the building?
solution
In the sketch here, the rightmost vertical line represents the building and the hypotenuse represents the
path of the laser beam. Because the laser pointer is 4◦
tilted from pointing straight up, the
b
2
86
20
angle formed by the laser beam and a line parallel to the ground is 86◦ ,
as indicated in the figure (which is not drawn to scale).
The side of the right triangle opposite the 86◦ angle has been labeled b.
Thus the height of the building is b + 2.
We have tan 86◦ =
b
20 .
Solving this equation for b, we get
b = 20 tan 86◦ ≈ 286.
Adding 2 to this result, we see that the height of the building is approximately 288 meters.
Instructor’s Solutions Manual, Section 5.6
Exercise 1
Solutions to Exercises, Section 5.6
1.
For θ = 7◦ , evaluate each of the following:
(a) cos2 θ
(b) cos(θ 2 )
[Exercises 1 and 2 emphasize that cos2 θ does not equal cos(θ 2 ).]
solution
(a) Using a calculator working in degrees, we have
cos2 7◦ = (cos 7◦ )2 ≈ (0.992546)2 ≈ 0.985148.
(b) Note that 72 = 49. Using a calculator working in degrees, we have
cos 49◦ ≈ 0.656059.
Student Solution Manual, Section 5.6
3.
Exercise 3
For θ = 4 radians, evaluate each of the following:
(a) sin2 θ
(b) sin(θ 2 )
[Exercises 3 and 4 emphasize that sin2 θ does not equal sin(θ 2 ).]
solution
(a) Using a calculator working in radians, we have
sin2 4 = (sin 4)2 ≈ (−0.756802)2 ≈ 0.57275.
(b) Note that 42 = 16. Using a calculator working in radians, we have
sin 16 ≈ −0.287903.
Instructor’s Solutions Manual, Section 5.6
Exercise 5
In Exercises 5–38, find exact expressions for the indicated quantities, given
that
√
√
2
+
3
2− 2
π
π
and sin 8 =
.
cos 12 =
2
2
[These values for cos
Section 6.3.]
π
12
and sin
π
8
will be derived in Examples 4 and 5 in
π
5. cos(− 12 )
solution
π
cos(− 12 )
= cos
π
12
√
2+ 3
=
2
Student Solution Manual, Section 5.6
Exercise 7
π
7. sin 12
solution We know that
π
π
cos2 12 + sin2 12 = 1.
Thus
π
π
sin2 12 = 1 − cos2 12
2 + √3 2
=1−
2
√
2+ 3
=1−
4
√
2− 3
.
=
4
π
Because sin 12 > 0, taking square roots of both sides of the equation
above gives
√
2− 3
π
.
sin 12 =
2
Instructor’s Solutions Manual, Section 5.6
Exercise 9
π
9. sin(− 12
)
solution
π
sin(− 12 )
= − sin
π
12
√
2− 3
=−
2
Student Solution Manual, Section 5.6
Exercise 11
π
11. tan 12
solution
π
tan 12 =
π
sin 12
π
cos 12
√
2− 3
=
√
2+ 3
√
√
2− 3
2− 3
= √ ·
√
2+ 3
2− 3
√
2− 3
= √
4−3
√
=2− 3
Instructor’s Solutions Manual, Section 5.6
Exercise 13
π
13. tan(− 12
)
solution
π
π
tan(− 12 ) = − tan 12 = −(2 −
√
3) =
√
3−2
Student Solution Manual, Section 5.6
Exercise 15
15. cos 25π
12
solution Because
25π
12
=
π
12
cos
+ 2π , we have
25π
12
π
= cos( 12 + 2π )
π
= cos 12
√
2+ 3
.
=
2
Instructor’s Solutions Manual, Section 5.6
Exercise 17
17. sin 25π
12
solution Because
25π
12
=
π
12
sin
+ 2π , we have
25π
12
π
= sin( 12 + 2π )
π
= sin 12
√
2− 3
.
=
2
Student Solution Manual, Section 5.6
Exercise 19
19. tan 25π
12
solution Because
25π
12
=
π
12
tan
+ 2π , we have
25π
12
π
= tan( 12 + 2π )
π
= tan 12
√
= 2 − 3.
Instructor’s Solutions Manual, Section 5.6
Exercise 21
21. cos 13π
12
solution Because
13π
12
=
π
12
cos
+ π , we have
13π
12
π
= cos( 12 + π )
π
= − cos 12
√
2+ 3
.
=−
2
Student Solution Manual, Section 5.6
Exercise 23
23. sin 13π
12
solution Because
13π
12
=
π
12
sin
+ π , we have
13π
12
π
= sin( 12 + π )
π
= − sin 12
√
2− 3
.
=−
2
Instructor’s Solutions Manual, Section 5.6
Exercise 25
25. tan 13π
12
solution Because
13π
12
=
π
12
tan
+ π , we have
13π
12
π
= tan( 12 + π )
π
= tan 12
√
= 2 − 3.
Student Solution Manual, Section 5.6
Exercise 27
27. cos 5π
12
solution
cos
5π
12
=
π
sin( 2
−
5π
12 )
= sin
π
12
√
2− 3
=
2
Instructor’s Solutions Manual, Section 5.6
Exercise 29
29. cos(− 5π
12 )
solution
5π
cos(− 12 )
= cos
5π
12
√
2− 3
=
2
Student Solution Manual, Section 5.6
Exercise 31
31. sin 5π
12
solution
sin
5π
12
=
π
cos( 2
−
5π
12 )
= cos
π
12
√
2+ 3
=
2
Instructor’s Solutions Manual, Section 5.6
Exercise 33
33. sin(− 5π
12 )
solution
5π
sin(− 12 )
= − sin
5π
12
√
2+ 3
=−
2
Student Solution Manual, Section 5.6
Exercise 35
35. tan 5π
12
solution
tan
5π
12
=
1
tan( π2
−
=
1
π
tan 12
=
1
√
2− 3
5π
12 )
√
2+ 3
1
√ ·
√
=
2− 3 2+ 3
√
2+ 3
=
4−3
√
=2+ 3
Instructor’s Solutions Manual, Section 5.6
Exercise 37
37. tan(− 5π
12 )
solution
5π
tan(− 12 ) = − tan
5π
12
= −2 −
√
3
Student Solution Manual, Section 5.6
Exercise 39
Suppose u and ν are in the interval ( π
2 , π), with
tan u = −2 and
tan ν = −3.
In Exercises 39–66, find exact expressions for the indicated quantities.
39. tan(−u)
solution tan(−u) = − tan u = −(−2) = 2
Instructor’s Solutions Manual, Section 5.6
Exercise 41
41. cos u
solution We know that
−2 = tan u
=
sin u
.
cos u
√
To find cos u, make the substitution sin u = 1 − cos2 u in the equation
π
above (this substitution is valid because 2 < u < π , which implies that
sin u > 0), getting
√
1 − cos2 u
.
−2 =
cos u
Now square both sides of the equation above, then multiply both sides
by cos2 u and rearrange to get the equation
5 cos2 u = 1.
1
1
Thus cos u = − √5 (the possibility that cos u equals √5 is eliminated
π
because 2 < u < π , which implies that cos u < 0). This can be written
as cos u = −
√
5
5 .
Student Solution Manual, Section 5.6
43. cos(−u)
√
5
solution cos(−u) = cos u = −
5
Exercise 43
Instructor’s Solutions Manual, Section 5.6
45. sin u
solution
sin u = 1 − cos2 u
1
= 1−
5
4
=
5
2
= √
5
√
2 5
=
5
Exercise 45
Student Solution Manual, Section 5.6
47. sin(−u)
solution
√
2 5
sin(−u) = − sin u = −
5
Exercise 47
Instructor’s Solutions Manual, Section 5.6
49. cos(u + 4π )
solution
√
5
cos(u + 4π ) = cos u = −
5
Exercise 49
Student Solution Manual, Section 5.6
51. sin(u − 6π )
solution
√
2 5
sin(u − 6π ) = sin u =
5
Exercise 51
Instructor’s Solutions Manual, Section 5.6
53. tan(u + 8π )
solution tan(u + 8π ) = tan u = −2
Exercise 53
Student Solution Manual, Section 5.6
55. cos(u − 3π )
solution
√
5
cos(u − 3π ) = − cos u =
5
Exercise 55
Instructor’s Solutions Manual, Section 5.6
57. sin(u + 5π )
solution
√
2 5
sin(u + 5π ) = − sin u = −
5
Exercise 57
Student Solution Manual, Section 5.6
59. tan(u − 9π )
solution tan(u − 9π ) = tan u = −2
Exercise 59
Instructor’s Solutions Manual, Section 5.6
61. cos( π2 − u)
solution
π
cos( 2
√
2 5
− u) = sin u =
5
Exercise 61
Student Solution Manual, Section 5.6
Exercise 63
63. sin( π2 − u)
solution
sin
π
2
√
5
− u = cos u = −
5
Instructor’s Solutions Manual, Section 5.6
Exercise 65
65. tan( π2 − u)
solution
π
tan( 2 − u) =
1
1
=−
tan u
2
Student Solution Manual, Section 5.7
Exercise 1
Solutions to Exercises, Section 5.7
1. Evaluate cos−1 21 .
solution cos
π
3
= 12 ; thus cos−1
1
2
=
π
3.
Instructor’s Solutions Manual, Section 5.7
3. Evaluate tan−1 (−1).
π
π
solution tan(− 4 ) = −1; thus tan−1 (−1) = − 4 .
Exercise 3
Student Solution Manual, Section 5.7
Exercise 5
c
Ν
b
u
a
Use the right triangle above for Exercises 5–12. This triangle is not drawn
to scale corresponding to the data in the exercises.
5.
Suppose a = 2 and c = 3. Evaluate u in radians.
solution Because the cosine of an angle in a right triangle equals the
length of the adjacent side divided by the length of the hypotenuse, we
have cos u = 23 . Using a calculator working in radians, we then have
u = cos−1
2
3
≈ 0.841 radians.
Instructor’s Solutions Manual, Section 5.7
7.
Exercise 7
Suppose a = 2 and c = 5. Evaluate ν in radians.
solution Because the sine of an angle in a right triangle equals the
length of the opposite side divided by the length of the hypotenuse, we
2
have sin ν = 5 . Using a calculator working in radians, we then have
ν = sin−1
2
5
≈ 0.412 radians.
Student Solution Manual, Section 5.7
9.
Exercise 9
Suppose a = 5 and b = 4. Evaluate u in degrees.
solution Because the tangent of an angle in a right triangle equals the
length of the opposite side divided by the length of the adjacent side, we
4
have tan u = 5 . Using a calculator working in degrees, we then have
u = tan−1
4
5
≈ 38.7◦ .
Instructor’s Solutions Manual, Section 5.7
11.
Exercise 11
Suppose a = 5 and b = 7. Evaluate ν in degrees.
solution Because the tangent of an angle in a right triangle equals the
length of the opposite side divided by the length of the adjacent side, we
5
have tan ν = 7 . Using a calculator working in degrees, we then have
ν = tan−1
5
7
≈ 35.5◦ .
Student Solution Manual, Section 5.7
13.
Exercise 13
Find the smallest positive number t such that 10cos t = 6.
solution The equation above implies that cos t = log 6. Thus we take
t = cos−1 (log 6) ≈ 0.67908.
Instructor’s Solutions Manual, Section 5.7
15.
Exercise 15
Find the smallest positive number t such that etan t = 15.
solution The equation above implies that tan t = ln 15. Thus we take
t = tan−1 (ln 15) ≈ 1.21706.
Student Solution Manual, Section 5.7
17.
Exercise 17
Find the smallest positive number y such that cos(tan y) = 0.2.
solution The equation above implies that we should choose tan y =
cos−1 0.2 ≈ 1.36944. Thus we should choose y ≈ tan−1 1.36944 ≈
0.94007.
Instructor’s Solutions Manual, Section 5.7
19.
Exercise 19
Find the smallest positive number x such that
sin2 x − 3 sin x + 1 = 0.
solution
as
Write y = sin x. Then the equation above can be rewritten
y 2 − 3y + 1 = 0.
Using the quadratic formula, we find that the solutions to this equation
are
√
3+ 5
≈ 2.61803
y=
2
and
√
3− 5
≈ 0.38197.
y=
2
Thus sin x ≈ 2.61803 or sin x ≈ 0.381966. However, there is no real
number x such that sin x ≈ 2.61803 (because sin x is at most 1 for
every real number x), and thus we must have sin x ≈ 0.381966. Thus
x ≈ sin−1 0.381966 ≈ 0.39192.
Student Solution Manual, Section 5.7
21.
Exercise 21
Find the smallest positive number x such that
cos2 x − 0.5 cos x + 0.06 = 0.
solution
as
Write y = cos x. Then the equation above can be rewritten
y 2 − 0.5y + 0.06 = 0.
Using the quadratic formula or factorization, we find that the solutions
to this equation are
y = 0.2 and y = 0.3.
Thus cos x = 0.2 or cos x = 0.3, which suggests that we choose x =
cos−1 0.2 or x = cos−1 0.3. Because arccosine is a decreasing function, cos−1 0.3 is smaller than cos−1 0.2. Because we want to find the
smallest positive value of x satisfying the original equation, we choose
x = cos−1 0.3 ≈ 1.2661.
Instructor’s Solutions Manual, Section 5.8
Exercise 1
Solutions to Exercises, Section 5.8
1. Suppose t is such that cos−1 t = 2. Evaluate the following:
(c) sin−1 (−t)
(a) cos−1 (−t)
−1
(b) sin
t
solution
(a) cos−1 (−t) = π − cos−1 t = π − 2
(b) sin−1 t =
−1
(c) sin
π
2
− cos−1 t =
−1
(−t) = − sin
π
2
−2
t =2−
π
2
Student Solution Manual, Section 5.8
Exercise 3
3. Suppose t is such that tan−1 t =
(a) tan−1
3π
7 .
Evaluate the following:
(c) tan−1 (− t )
1
1
t
(b) tan−1 (−t)
solution
(a) Because t = tan
3π
7 ,
we see that t > 0. Thus
tan−1
1
t
=
π
2
− tan−1 t =
3π
(b) tan−1 (−t) = − tan−1 t = − 7
π
(c) tan−1 (− 1t ) = − tan−1 ( 1t ) = − 14
π
2
−
3π
7
=
π
14 .
Instructor’s Solutions Manual, Section 5.8
Exercise 5
5. Evaluate cos(cos−1 41 ).
1
solution Let θ = cos−1 4 . Thus θ is the angle in [0, π ] such that
1
1
1
cos θ = 4 . Thus cos(cos−1 4 ) = cos θ = 4 .
Student Solution Manual, Section 5.8
Exercise 7
7. Evaluate sin−1 (sin 2π
7 ).
solution Let θ = sin−1 (sin
π π
interval [− 2 , 2 ] such that
2π
7 ).
Thus θ is the unique angle in the
sin θ = sin
1
2
1
Because − 2 ≤ 7 ≤ 2 , we see that
2π
above implies that θ = 7 .
2π
7
2π
7 .
π
is in [− 2 ,
π
2 ].
Thus the equation
Instructor’s Solutions Manual, Section 5.8
Exercise 9
9. Evaluate cos−1 (cos 3π ).
solution Because cos 3π = −1, we see that
cos−1 (cos 3π ) = cos−1 (−1).
Because cos π = −1, we have cos−1 (−1) = π (cos 3π also equals −1, but
cos−1 (−1) must be in the interval [0, π ]). Thus cos−1 (cos 3π ) = π .
Student Solution Manual, Section 5.8
Exercise 11
11. Evaluate tan−1 (tan 11π
5 ).
solution Because tan−1 is the inverse of tan, it may be tempting to think
11π
11π
that tan−1 (tan 5 ) equals 5 . However, the values of tan−1 must be
π
π
11π
π
11π
between − 2 and 2 . Because 5 > 2 , we conclude that tan−1 (tan 5 )
11π
cannot equal 5 .
Note that
tan 11π
5 = tan(2π +
Because
π
5
π
is in (− 2 ,
π
2 ),
π
5)
= tan π5 .
we have tan−1 (tan
tan−1 (tan
11π
5 )
π
5)
=
π
5.
= tan−1 (tan π5 ) =
Thus
π
5.
Instructor’s Solutions Manual, Section 5.8
13. Evaluate sin(− sin−1
Exercise 13
3
13 ).
solution
sin(− sin−1
3
13 )
= − sin(sin−1
3
= − 13
3
13 )
Student Solution Manual, Section 5.8
Exercise 15
15. Evaluate sin(cos−1 31 ).
solution We give two ways to work this exercise: the algebraic approach and the right-triangle approach.
1
Algebraic approach: Let θ = cos−1 3 . Thus θ is the angle in [0, π ] such
1
that cos θ = 3 . Note that sin θ ≥ 0 because θ is in [0, π ]. Thus
sin(cos−1 3 ) = sin θ
= 1 − cos2 θ
1
= 1− 9
8
= 9
1
=
√
2 2
3 .
1
1
Right-triangle approach: Let θ = cos−1 3 ; thus cos θ = 3 . Because
cos θ =
adjacent side
hypotenuse
in a right triangle with an angle of θ, the following figure (which is not
drawn to scale) illustrates the situation:
Instructor’s Solutions Manual, Section 5.8
Exercise 15
3
Θ
b
1
We need to evaluate sin θ. In terms of the figure above, we have
sin θ =
b
opposite side
= .
hypotenuse
3
Applying the Pythagorean Theorem to the triangle above,
we have b2 +1 =
√
√
√
2 2
9, which implies that b = 8 = 2 2. Thus sin θ = 3 . In other words,
1
sin(cos−1 3 ) =
√
2 2
3 .
Student Solution Manual, Section 5.8
Exercise 17
17. Evaluate tan(cos−1 31 ).
solution We give two ways to work this exercise: the algebraic approach and the right-triangle approach.
Algebraic approach: From Exercise 15, we already know that
sin(cos−1 3 ) =
1
√
2 2
3 .
Thus
tan(cos−1 3 ) =
1
=
sin(cos−1 31 )
1
cos(cos−1 3 )
√
2 2
3
1
3
√
= 2 2.
1
1
Right-triangle approach: Let θ = cos−1 3 ; thus cos θ = 3 . Because
cos θ =
adjacent side
hypotenuse
in a right triangle with an angle of θ, the following figure (which is not
drawn to scale) illustrates the situation:
Instructor’s Solutions Manual, Section 5.8
Exercise 17
3
Θ
b
1
We need to evaluate tan θ. In terms of the figure above, we have
tan θ =
opposite side
= b.
adjacent side
Applying the Pythagorean Theorem to the triangle above, we have b2 +1 =
√
√
√
9, which implies that b = 8 = 2 2. Thus tan θ = 2 2. In other words,
√
1
tan(cos−1 3 ) = 2 2.
Student Solution Manual, Section 5.8
Exercise 19
19. Evaluate cos tan−1 (−4) .
solution We give two ways to work this exercise: the algebraic approach and the right-triangle approach.
π π
Algebraic approach: Let θ = tan−1 (−4). Thus θ is the angle in (− 2 , 2 )
π π
such that tan θ = −4. Note that cos θ > 0 because θ is in (− 2 , 2 ).
Recall that dividing both sides of the identity cos2 θ + sin2 θ = 1 by cos2 θ
1
produces the equation 1 + tan2 θ = cos2 θ . Solving this equation for cos θ
gives the following:
cos tan−1 (−4) = cos θ
1
= 1 + tan2 θ
1
= 1 + (−4)2
=
√1
17
=
√
17
17 .
Right-triangle approach: Sides with a negative length make no sense in
a right triangle. Thus first we use some identities to get rid of the minus
sign, as follows:
cos tan−1 (−4) = cos(− tan−1 4)
= cos(tan−1 4).
Instructor’s Solutions Manual, Section 5.8
Exercise 19
Thus we need to evaluate cos(tan−1 4).
Now let θ = tan−1 4; thus tan θ = 4. Because
tan θ =
opposite side
adjacent side
in a right triangle with an angle of θ, the following figure (which is not
drawn to scale) illustrates the situation:
c
Θ
4
1
We need to evaluate cos θ. In terms of the figure above, we have
cos θ =
1
adjacent side
= .
hypotenuse
c
2
Applying the Pythagorean Theorem to the triangle above, we
√ have c =
√
1
17
1 + 16, which implies that c = 17. Thus cos θ = √17 = 17 . In other
√
√17
17
words, cos(tan−1 4) = 17 . Thus cos tan−1 (−4) = 17 .
Student Solution Manual, Section 5.8
Exercise 21
21. Evaluate sin−1 (cos 2π
5 ).
solution
sin−1 (cos
2π
5 )
=
π
2
− cos−1 (cos
=
π
2
−
2π
5
=
π
10
2π
5 )
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