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**13-3 The Unit Circle Warm Up Lesson Presentation Lesson Quiz**

Holt Algebra 2

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**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

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**Objectives Convert angle measures between degrees and radians.**

Find the values of trigonometric functions on the unit circle.

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Vocabulary radian unit circle

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**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.

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

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.

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**Example 1: Converting Between Degrees and Radians**

Convert each measure from degrees to radians or from radians to degrees. A. – 60° . B.

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**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

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

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

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**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:

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**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.

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**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.

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**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 = .

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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.

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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 = .

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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.

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

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

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**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°

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Example 3 Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x.

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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.

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**Step 1 Find the measure of the reference angle.**

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°

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**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.

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**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.

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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 .

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**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.

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**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.

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**Step 1 Find the measure of the reference angle.**

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°.

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**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.

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**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.

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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.

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**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.

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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.

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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.

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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.

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**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.

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**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.

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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

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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

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