Unit Circle, Arcs, and Sectors
Unit Circle, Arcs, and Sectors
Objectives:
1.
To complete and use the unit circle
2.
To find circumference and arc length
3.
To find the area of circles and sectors
Assignment:
•
Complete Unit Circle
•
Challenge Problems
Objective 1
You will be able to complete and use the unit circle
Exercise 1
Solve each right triangle. Write your answers in simplest radical form.
1 unit
1 unit
30
45
Radians
Radians are another way to measure an angle. If you take the radius and wrap it around the circle, the angle that is formed is one
radian
.
Radians
It takes a little bit more than 3 radians to span a semicircle.
That “little bit more than 3” is π.
So π radians = 180° and 2π radians =
360°
Exercise 2
Rewrite each of the following angle measures in terms of radians.
(180
° = π rad)
1.
30
°
2.
45
°
3.
60
°
4.
90
°
Exercise 3
Write the equation of the circle.
The Unit Circle
This tiny circle is called the unit circle since its radius is 1 unit. This circle may be tiny, but it will give us a way to find 102 exact trig values.
That’s pretty useful.
Unit Circle Activity
Math students often use a
unit circle
to find the exact trig ratios of certain angle measures, since most calculators won’t divulge that information. In this activity, we will construct a unit circle.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
The outer bold circle is the unit circle.
We will eventually be labeling each of the points along this bold circle with ordered pairs.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
Now look at the inner circle. The points along this circle will be labeled with degree measures. What do you suppose these degree measures represent?
Finish the degree measures.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
Now look at the middle circle. The points along this circle will be labeled with radian measures.
Finish the radian measures for the first quadrant.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
Finally, look at the outer circle again. Let’s concentrate on the point that is 30
° along this circle.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
Finally, look at the outer circle again. Let’s concentrate on the point that is 30
° along this circle.
Realize that the ordered pair for any point makes a right triangle with the xaxis.
y
90
60
45
30
0
360
0
2
x
Unit Circle Activity
1.
What is the length of the hypotenuse?
2.
What is the length of the short leg?
3.
What is the length of the longer leg?
4.
What are the coordinates of the point on the circle?
y
90
60
45
30
0
360
0
2
2
,
1
2
x
Unit Circle Activity
How can this unit circle be used to find the following?
1.
cos (30
°)
2.
sin (30
°)
3.
tan (30
°)
y
(0, 1)
90
60
45
30
1
0
360
2
3
0
2
1
2
2
,
1
2
(1, 0)
x
Unit Circle Activity
Let’s look at the outer circle again. This time concentrate on the point that is 45
° along this circle.
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
1.
What is the length of the hypotenuse?
2.
What is the length of the base leg?
3.
What is the length of the height leg?
4.
What are the coordinates of the point on the circle?
90
60
45
30
0
360
2
2
2
,
2
2
Unit Circle Activity
How can this unit circle be used to find the following?
1.
cos (45
°)
2.
sin (45
°)
3.
tan (45
°)
y
(0, 1)
90
1
60
45
30
0
360
2
2
0
2
2
2
2
2
,
2
2
(1, 0)
x
Unit Circle Activity
In general, for any point (x, y) along the outer circle of the unit circle:
1.
cos(
) = x
2.
sin (
) = y
3.
tan (
) = y/x
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
Let’s look at the outer circle one last time.
This time let’s look at the point that is 150
° along the unit circle.
Obviously, we cannot make a right triangle with a 150
° angle. So how could we complete the second quadrant?
y
(0, 1)
90
60
45
30
0
360
0
2
(1, 0)
x
Unit Circle Activity
The answer involves symmetry.
y

2
2
,
2
2

2
3
, , ,
1
2
90
60
45
30
0
360
0
2
, ,
x
Unit Circle Activity
The same would apply for the 3 rd and 4 th quadrants.
y
(0, 1)

2
2
,
2
2

3
2
,
1
2
2
2
,
2
2
2
,
1
2
90
60
45
30
0
360
0
2
(1, 0)
x
Objective 2
You will be able to find circumference and arc length
Exercise 4
A tennis ball can is approximately 3 tennis ball diameters high, and its width is approximately one tennis ball diameter.
Which is greater: the height of a tennis ball can or the distance around the can?
Let’s Have Some π
The distance around a circle is its
circumference
.
Diameter
Circumference
Approximating Pi
Archimedes of Syracuse first approximated pi as 3.14 using the perimeters of inscribed and circumscribed polygons.
Circumference of a Circle
The circumference C of a circle is C = πd or
C = 2πr, where d is the diameter and r is the radius of the circle.
Arc Measure and Arc Length
The
measure of an arc
is the measure of the central angle it intercepts. It is measured in degrees.
Arc Measure and Arc Length
An arc length is a portion of the circumference of a circle. It is measured in linear units and can be found using the measure of the arc.
Exercise 5
Assume the radius of each circle below is 24 units. Find the length of arc AB.
B
B
B
90
60
45
A
A
A
Arc Length Corollary
If the radius of a circle is 𝑟 and two radii form a central angle of 𝑎°, then the length of the arc formed by those radii is given by the formula: r a
B
A
Arc Length Corollary
If the radius of a circle is 𝑟 and two radii form a central angle of 𝑎°, then the length of the arc formed by those radii is given by the formula: r a
B
A
Exercise 6
Find the length of arc AB.
1.
2.
3.
Exercise 7
Find the indicated measure.
1. Circumference = 2. Radius =
Exercise 8
Find the perimeter of the region.
1.
2.
Objective 3
You will be able to find the area of circles and sectors
Area of a Circle
Recall that when investigating the area of certain polygons, we based our new area formulas on shapes whose area we already knew. We’ll do the same with the area of a circle. (It’s probably the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain polygons, we based our new area formulas on shapes whose area we already knew. We’ll do the same with the area of a circle. (It’s probably the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain polygons, we based our new area formulas on shapes whose area we already knew. We’ll do the same with the area of a circle. (It’s probably the coolest thing you’ve ever seen.)
Area of a Circle
Area of a Circle Theorem
The area of a circle is
π times the square of the radius.
Sector: An Actual Slice of Pie
A sector of a circle is the region between two radii of a circle and the included arc.
Exercise 9
Remember that delicious cherry pie from Example
5? Well, it’s mostly a memory now, as there is only one slice left! What is the area of that piece of pie if the radius is 6 inches and the angle formed by the sides of the pie slice is 60°?
Got any vanilla ice cream for that?
Area of a Sector Conjecture
If the radius of a circle is r and two radii form a central angle of a°, then the area of the sector formed by those radii is given by the formula
a
360
r
2
Area of a Sector Conjecture
If the radius of a circle is r and two radii form a central angle of a°, then the area of the sector formed by those radii is given by the formula
Exercise 10
The radius of a circle is 18 cm. A sector is formed by a central angle measuring 40°.
What is the exact area of the sector?
Exercise 11
Find the area of each sector.
1.
2.
Exercise 12
1.
Find the area of circle S.
2.
Find the radius of circle S.
Exercise 13
What is the area of the shaded region if the radius of each circle is 6 cm.
Unit Circle, Arcs, and Sectors
Objectives:
1.
To complete and use the unit circle
2.
To find circumference and arc length
3.
To find the area of circles and sectors
Assignment:
•
Complete Unit Circle
•
Challenge Problems
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