The Unit Circle

1
13-3
The Unit Circle
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up Find the measure of the reference angle
for each given angle. 1. 120 2. 225 3.
150 4. 315 Find the exact value of each
trigonometric function. 5. sin 60 6. tan 45
7. cos 45 8. cos 60
60
45
30
45
1
3
Objectives
Convert angle measures between degrees and
radians. Find the values of trigonometric
functions on the unit circle.
4
Vocabulary
radian unit circle
5
So far, you have measured angles in degrees. You
can also measure angles in radians.
A radian is a unit of angle measure based on arc
length. Recall from geometry that an arc is an
unbroken part of a circle. If a central angle ?
in a circle of radius r, then the measure of ? is
defined as 1 radian.
6
The circumference of a circle of radius r is 2?r.
Therefore, an angle representing one complete
clockwise rotation measures 2? radians. You can
use the fact that 2? radians is equivalent to
360 to convert between radians and degrees.
7
(No Transcript)
8
Example 1 Converting Between Degrees and Radians
Convert each measure from degrees to radians or
from radians to degrees.
A. 60
B.
9
(No Transcript)
10
Check It Out! Example 1
Convert each measure from degrees to radians or
from radians to degrees.
a. 80
b.
11
Check It Out! Example 1
Convert each measure from degrees to radians or
from radians to degrees.
c. 36
d. 4? radians
12
A unit circle is a circle with a radius of 1
unit. For every point P(x, y) on the unit circle,
the value of r is 1. Therefore, for an angle ? in
the standard position
13
So the coordinates of P can be written as (cos?,
sin?).
The diagram shows the equivalent degree and
radian measure of special angles, as well as the
corresponding x- and y-coordinates of points on
the unit circle.
14
Example 2A Using the Unit Circle to Evaluate
Trigonometric Functions
Use the unit circle to find the exact value of
each trigonometric function.
cos 225
cos 225 x
Use cos ? x.
15
Example 2B Using the Unit Circle to Evaluate
Trigonometric Functions
Use the unit circle to find the exact value of
each trigonometric function.
tan
16
Check It Out! Example 1a
Use the unit circle to find the exact value of
each trigonometric function.
sin 315
sin 315 y
Use sin ? y.
17
Check It Out! Example 1b
Use the unit circle to find the exact value of
each trigonometric function.
tan 180
The angle passes through the point (1, 0) on the
unit circle.
18
Check It Out! Example 1c
Use the unit circle to find the exact value of
each trigonometric function.
19
You can use reference angles and Quadrant I of
the unit circle to determine the values of
trigonometric functions.
20
The diagram shows how the signs of the
trigonometric functions depend on the quadrant
containing the terminal side of ? in standard
position.
21
Example 3 Using Reference Angles to Evaluate
Trigonometric functions
Use a reference angle to find the exact value of
the sine, cosine, and tangent of 330.
Step 1 Find the measure of the reference angle.
The reference angle measures 30
22
Example 3 Continued
Step 2 Find the sine, cosine, and tangent of the
reference angle.
Use sin ? y.
Use cos ? x.
23
Example 3 Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin ? is negative.
In Quadrant IV, cos ? is positive.
In Quadrant IV, tan ? is negative.
24
Check It Out! Example 3a
Use a reference angle to find the exact value of
the sine, cosine, and tangent of 270.
Step 1 Find the measure of the reference angle.
The reference angle measures 90
25
Check It Out! Example 3a Continued
Step 2 Find the sine, cosine, and tangent of the
reference angle.
Use sin ? y.
sin 90 1
Use cos ? x.
cos 90 0
tan 90 undef.
26
Check It Out! Example 3a Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin ? is negative.
sin 270 1
cos 270 0
tan 270 undef.
27
Check It Out! Example 3b
Use a reference angle to find the exact value of
the sine, cosine, and tangent of each angle.
Step 1 Find the measure of the reference angle.
28
Check It Out! Example 3b Continued
Step 2 Find the sine, cosine, and tangent of the
reference angle.
Use sin ? y.
Use cos ? x.
29
Check It Out! Example 3b Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin ? is negative.
In Quadrant IV, cos ? is positive.
In Quadrant IV, tan ? is negative.
30
Check It Out! Example 3c
Use a reference angle to find the exact value of
the sine, cosine, and tangent of each angle.
30
Step 1 Find the measure of the reference angle.
The reference angle measures 30.
31
Check It Out! Example 3c Continued
Step 2 Find the sine, cosine, and tangent of the
reference angle.
Use sin ? y.
Use cos ? x.
32
Check It Out! Example 3c Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin ? is negative.
In Quadrant IV, cos ? is positive.
In Quadrant IV, tan ? is negative.
33
If you know the measure of a central angle of a
circle, you can determine the length s of the arc
intercepted by the angle.
34
(No Transcript)
35
Example 4 Automobile Application
A tire of a car makes 653 complete rotations in 1
min. The diameter of the tire is 0.65 m. To the
nearest meter, how far does the car travel in 1 s?
Step 1 Find the radius of the tire.
Step 2 Find the angle ? through which the tire
rotates in 1 second.
Write a proportion.
36
Example 4 Continued
The tire rotates ? radians in 1 s and 653(2?)
radians in 60 s.
Cross multiply.
Divide both sides by 60.
Simplify.
37
Example 4 Continued
Use the arc length formula.
Simplify by using a calculator.
The car travels about 22 meters in second.
38
Check It Out! Example 4
An minute hand on Big Bens Clock Tower in London
is 14 ft long. To the nearest tenth of a foot,
how far does the tip of the minute hand travel in
1 minute?
Step 1 Find the radius of the clock.
The radius is the actual length of the hour hand.
r 14
Step 2 Find the angle ? through which the hour
hand rotates in 1 minute.
Write a proportion.
39
Check It Out! Example 4 Continued
The hand rotates ? radians in 1 m and 2? radians
in 60 m.
Cross multiply.
Divide both sides by 60.
Simplify.
40
Check It Out! Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
s 1.5 feet
Simplify by using a calculator.
The minute hand travels about 1.5 feet in one
minute.
41
Lesson Quiz Part I
Convert each measure from degrees to radians or
from radians to degrees.
2.
144
1. 100
4. Use a reference angle to find the exact value
of the sine, cosine, and tangent of
42
Lesson Quiz Part II
5. A carpenter is designing a curved piece of
molding for the ceiling of a museum. The curve
will be an arc of a circle with a radius of 3 m.
The central angle will measure 120. To the
nearest tenth of a meter, what will be the length
of the molding?
6.3 m