1. 13-3 The Unit Circle Warm Up Lesson Presentation Lesson Quiz
Holt Algebra 2

2. Find the measure of the reference angle for each given angle.
Warm Up
Find the measure of the reference angle for each given angle.
1. 120° °
3. –150° °
Find the exact value of each trigonometric function.
5. sin 60° 6. tan 45°
7. cos 45° 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
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 θ in a circle of radius r, then the measure of θ is defined as 1 radian.

6. The circumference of a circle of radius r is 2r
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.

8. Example 1: Converting Between Degrees and Radians
Convert each measure from degrees to radians or from radians to degrees.
A. – 60° . B.

9. Angles measured in radians are often not labeled with the unit
Angles measured in radians are often not labeled with the unit. If an angle measure does not have a degree symbol, you can usually assume that the angle is measured in radians.
Reading Math

10. Check It Out! Example 1
Convert each measure from degrees to radians or from radians to degrees.
a. 80° 4 9 . b. 20 .

11. Check It Out! Example 1
Convert each measure from degrees to radians or from radians to degrees.
c. –36° 5 .
d. 4 radians .

12. A unit circle is a circle with a radius of 1 unit
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 θ in the standard position:

13. So the coordinates of P can be written as (cosθ, sinθ).
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. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?

15. Example 2A: Using the Unit Circle to Evaluate Trigonometric Functions
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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°
The angle passes through the point
on the unit circle.
cos 225° = x
Use cos θ = x.

16. Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each trigonometric function.
tan
The angle passes through the point
on the unit circle.
Use tan θ = .

17. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 1a
Use the unit circle to find the exact value of each trigonometric function.
sin 315°
The angle passes through the point
on the unit circle.
sin 315° = y
Use sin θ = y.

18. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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.
tan 180° =
Use tan θ = .

19. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 1c
Use the unit circle to find the exact value of each trigonometric function.
The angle passes through the point
on the unit circle.

20. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions.
Trigonometric Functions and Reference Angles

21. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

22. Example 3: Using Reference Angles to Evaluate Trigonometric functions
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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°

23. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Example 3 Continued
Step 2 Find the sine, cosine, and tangent of the reference angle.
Use sin θ = y.
Use cos θ = x.

24. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Example 3 Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is negative.
In Quadrant IV, cos θ is positive.
In Quadrant IV, tan θ is negative.

25. Step 1 Find the measure of the reference angle.
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3a
Use a reference angle to find the exact value of the sine, cosine, and tangent of 270°.
270°
Step 1 Find the measure of the reference angle.
The reference angle measures 90°

26. Check It Out! Example 3a Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3a Continued
Step 2 Find the sine, cosine, and tangent of the reference angle.
90°
sin 90° = 1
Use sin θ = y.
cos 90° = 0
Use cos θ = x.
tan 90° = undef.

27. Check It Out! Example 3a Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3a Continued
Step 3 Adjust the signs, if needed.
sin 270° = –1
In Quadrant IV, sin θ is negative.
cos 270° = 0
tan 270° = undef.

28. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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.
The reference angle measures .

29. Check It Out! Example 3b Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3b Continued
Step 2 Find the sine, cosine, and tangent of the reference angle.
30°
Use sin θ = y.
Use cos θ = x.

30. Check It Out! Example 3b Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3b Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is negative.
In Quadrant IV, cos θ is positive.
In Quadrant IV, tan θ is negative.

31. Step 1 Find the measure of the reference angle.
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3c
Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.
–30°
–30°
Step 1 Find the measure of the reference angle.
The reference angle measures 30°.

32. Check It Out! Example 3c Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3c Continued
Step 2 Find the sine, cosine, and tangent of the reference angle.
30°
Use sin θ = y.
Use cos θ = x.

33. Check It Out! Example 3c Continued
EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Check It Out! Example 3c Continued
Step 3 Adjust the signs, if needed.
In Quadrant IV, sin θ is negative.
In Quadrant IV, cos θ is positive.
In Quadrant IV, tan θ is negative.

34. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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.

35. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?

36. 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.
The radius is of the diameter.
Step 2 Find the angle θ through which the tire rotates in 1 second.
Write a proportion.

37. Example 4 Continued
The tire rotates θ radians in 1 s and 653(2) radians in 60 s.
Cross multiply.
Divide both sides by 60.
Simplify.

38. Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
Substitute for r and for θ
Simplify by using a calculator.
The car travels about 22 meters in second.

39. Check It Out! Example 4
An minute hand on Big Ben’s 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 θ through which the hour hand rotates in 1 minute.
Write a proportion.

40. Check It Out! Example 4 Continued
The hand rotates θ radians in 1 m and 2 radians in 60 m.
Cross multiply.
Divide both sides by 60.
Simplify.

41. Check It Out! Example 4 Continued
Step 3 Find the length of the arc intercepted by
radians.
Use the arc length formula.
Substitute 14 for r and for θ.
s ≈ 1.5 feet
Simplify by using a calculator.
The minute hand travels about 1.5 feet in one minute.

42. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
Lesson Quiz: Part I
Convert each measure from degrees to radians or from radians to degrees.
1. 100° 2.
144°
3. Use the unit circle to find the exact value of
4. Use a reference angle to find the exact value of the sine, cosine, and tangent of

43. EQ: How can we use the Unit Circle to help us solve for exact values of the sine, cosine, and tangent?
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