13-3

Holt Algebra 2

13-3 The Unit Circle13-3 The Unit Circle

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2

13-3 The Unit Circle

Warm UpFind 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

Holt Algebra 2

13-3 The Unit Circle

Convert angle measures between degrees and radians.

Find the values of trigonometric functions on the unit circle.

Objectives

Holt Algebra 2

13-3 The Unit Circle

radianunit circle

Vocabulary

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

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

Holt Algebra 2

13-3 The Unit Circle

Holt Algebra 2

13-3 The Unit Circle

Example 1: Converting Between Degrees and Radians

Convert each measure from degrees to radians or from radians to degrees.

A. – 60°

B.

.

Holt Algebra 2

13-3 The Unit Circle

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

Holt Algebra 2

13-3 The Unit Circle

Check It Out! Example 1

Convert each measure from degrees to radians or from radians to degrees.

a. 80°

b.

.

.

4

9

20

Holt Algebra 2

13-3 The Unit Circle

Check It Out! Example 1

Convert each measure from degrees to radians or from radians to degrees.

c. –36°

d. 4 radians.

.

5

Holt Algebra 2

13-3 The Unit Circle

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:

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

tan

Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions

Use the unit circle to find the exact value of each trigonometric function.

The angle passes through the point

on the unit circle.

Use tan θ = .

Holt Algebra 2

13-3 The Unit Circle

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 θ = y.

The angle passes through the point

on the unit circle.

Holt Algebra 2

13-3 The Unit Circle

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 = .

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions.

Trigonometric Functions and Reference Angles

Holt Algebra 2

13-3 The Unit Circle

The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

Holt Algebra 2

13-3 The Unit Circle

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°

Holt Algebra 2

13-3 The Unit Circle

Example 3 Continued

Step 2 Find the sine, cosine, and tangent of the reference angle.

Use sin θ = y.

Use cos θ = x.

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

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°

270°

Holt Algebra 2

13-3 The Unit Circle

Step 2 Find the sine, cosine, and tangent of the reference angle.

Use sin θ = y.

Use cos θ = x.

Check It Out! Example 3a Continued

90°

tan 90° = undef.

sin 90° = 1

cos 90° = 0

Holt Algebra 2

13-3 The Unit Circle

Step 3 Adjust the signs, if needed.

In Quadrant IV, sin θ is negative.

Check It Out! Example 3a Continued

sin 270° = –1

cos 270° = 0

tan 270° = undef.

Holt Algebra 2

13-3 The Unit Circle

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 .

Holt Algebra 2

13-3 The Unit Circle

Check It Out! Example 3b Continued

Step 2 Find the sine, cosine, and tangent of the reference angle.

Use sin θ = y.

Use cos θ = x.

30°

Holt Algebra 2

13-3 The Unit Circle

Step 3 Adjust the signs, if needed.

In Quadrant IV, sin θ is negative.

Check It Out! Example 3b Continued

In Quadrant IV, cos θ is positive.

In Quadrant IV, tan θ is negative.

Holt Algebra 2

13-3 The Unit Circle

Check It Out! Example 3c

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 30°.

–30°

–30°

Holt Algebra 2

13-3 The Unit Circle

Check It Out! Example 3c Continued

Step 2 Find the sine, cosine, and tangent of the reference angle.

Use sin θ = y.

Use cos θ = x.

30°

Holt Algebra 2

13-3 The Unit Circle

Step 3 Adjust the signs, if needed.

In Quadrant IV, sin θ is negative.

Check It Out! Example 3c Continued

In Quadrant IV, cos θ is positive.

In Quadrant IV, tan θ is negative.

Holt Algebra 2

13-3 The Unit Circle

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.

Holt Algebra 2

13-3 The Unit Circle

Holt Algebra 2

13-3 The Unit Circle

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 θ through which the tire rotates in 1 second.

The radius is of the diameter.

Write a proportion.

Holt Algebra 2

13-3 The Unit Circle

Example 4 Continued

The tire rotates θ radians in 1 s and 653(2) radians in 60 s.

Simplify.

Divide both sides by 60.

Cross multiply.

Holt Algebra 2

13-3 The Unit Circle

Example 4 Continued

Step 3 Find the length of the arc intercepted by radians.

Use the arc length formula.

Simplify by using a calculator.

Substitute 0.325 for r and for θ

The car travels about 22 meters in second.

Holt Algebra 2

13-3 The Unit Circle

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.

Step 2 Find the angle θ through which the hour hand rotates in 1 minute.

Write a proportion.

r =14

Holt Algebra 2

13-3 The Unit Circle

The hand rotates θ radians in 1 m and 2 radians in 60 m.

Simplify.

Divide both sides by 60.

Cross multiply.

Check It Out! Example 4 Continued

Holt Algebra 2

13-3 The Unit Circle

Step 3 Find the length of the arc intercepted by radians.

Use the arc length formula.

Simplify by using a calculator.

The minute hand travels about 1.5 feet in one minute.

Check It Out! Example 4 Continued

Substitute 14 for r and for θ.

s ≈ 1.5 feet

Holt Algebra 2

13-3 The Unit Circle

Lesson Quiz: Part I

Convert each measure from degrees to radians or from radians to degrees.

1. 100° 2.

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

144°

Holt Algebra 2

13-3 The Unit Circle

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