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

13-3 The Unit Circle

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

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

Convert angle measures between degrees and radians.

Find the values of trigonometric functions on the unit circle.

Objectives

radianunit circle

Vocabulary

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.

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.

Example 1: Converting Between Degrees and Radians

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

A. – 60°

B.

.

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

Check It Out! Example 1


a. 80°

b.

.

.

4

9

20



c. –36°

d. 4 radians.

.

5

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:

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.

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.


tan

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



on the unit circle.

Use tan θ = .


Check It Out! Example 1a


sin 315°

sin 315° = y Use sin θ = y.


on the unit circle.


Check It Out! Example 1b


tan 180°


(–1, 0) on the unit circle.

tan 180° = Use tan θ = .


Check It Out! Example 1c



on 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


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


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°


Example 3 Continued

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

Use sin θ = y.

Use cos θ = x.


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.


Check It Out! Example 3a

Use a reference angle to find the exact value of the sine, cosine, and tangent of 270°.


The reference angle measures 90°

270°



Use sin θ = y.

Use cos θ = x.

Check It Out! Example 3a Continued

90°

tan 90° = undef.

sin 90° = 1

cos 90° = 0




Check It Out! Example 3a Continued

sin 270° = –1

cos 270° = 0

tan 270° = undef.


Check It Out! Example 3b

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.


The reference angle measures .


Check It Out! Example 3b Continued


Use sin θ = y.

Use cos θ = x.

30°




Check It Out! Example 3b Continued




Check It Out! Example 3c

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.


The reference angle measures 30°.

–30°

–30°


Check It Out! Example 3c Continued


Use sin θ = y.

Use cos θ = x.

30°




Check It Out! Example 3c Continued




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.


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.

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.

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.


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

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

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

Lesson Quiz: Part I


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°


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


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

Documents

Transcript of 13-3 The Unit Circle Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson...