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Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225°

Convert angle measures between degrees and radians, find values of trigonometric functions on the unit circle. Learn how to use the unit circle and evaluate trigonometric functions like sine, cosine, and tangent in both degrees and radians.

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Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225°

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  1. 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

  2. Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle.

  3. 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.

  4. The circumference of a circle of radius r is 2r. You can use 2radians=360° to convert between radians and degrees.

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

  6. Reading Math 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.

  7. 4 9 . 20 . Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. a. 80° b.

  8. 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 θin the standard position:

  9. 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.

  10. The angle passes through the point on 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° cos 225° = x Use cos θ = x.

  11. The angle passes through the point on the unit circle. Use tan θ = . Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. tan

  12. The angle passes through the point on 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.

  13. tan 180° = Use tan θ = . 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.

  14. The angle passes through the point on the unit circle. Check It Out! Example 1c Use the unit circle to find the exact value of each trigonometric function.

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