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The Unit Circle

The Unit Circle. Essential Questions. How do we convert angle measures between degrees and radians? How do we find the values of trigonometric functions on the unit circle?. Holt McDougal Algebra 2. Holt Algebra 2.

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The Unit Circle

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  1. The Unit Circle Essential Questions • How do we convert angle measures between degrees and radians? • How do we find the values of trigonometric functions on the unit circle? Holt McDougal Algebra 2 Holt Algebra 2

  2. 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 is r, then the measure of θ is defined as 1 radian.

  3. 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.

  4. . Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. 1. – 60° 2.

  5. . Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. 3. 80° 4.

  6. . Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. 5. –36° 6. 4radians

  7. 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.

  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. Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. 7. cos 225° Use cos θ = x. cos 225° = x

  11. Use tan θ = . 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.

  12. 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 sin θ = y. sin 315° = y

  13. tan 180° = Use tan θ = . 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 (–1, 0) on the unit circle.

  14. 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 cos θ = x.

  15. Lesson 10.3 Practice A

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