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The TheUnit UnitCircle Circle • How do we convert angle measures between degrees and radians? • How do we find the values of trigonometric functions on the unit circle? HoltMcDougal Algebra 2Algebra 2 Holt 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 is r, then the measure of θ is defined as 1 radian. Holt McDougal Algebra 2 The Unit Circle 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. Holt McDougal Algebra 2 The Unit Circle Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. . 1. – 60° 3 2. 60 Holt McDougal Algebra 2 o The Unit Circle Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. . 3. 80° 4 9 4. 20 Holt McDougal Algebra 2 o The Unit Circle Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. . 5. –36° 5 6. 4 radians Holt McDougal Algebra 2 The Unit Circle 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. Holt McDougal Algebra 2 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: y tan x Holt McDougal Algebra 2 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 xand y-coordinates of points on the unit circle. Holt McDougal Algebra 2 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° The angle passes through the point on the unit circle. Use cos θ = x. cos 225° = x Holt McDougal Algebra 2 The Unit Circle Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. 5 8. tan 6 The angle passes through the point on the unit circle. Use tan θ = Holt McDougal Algebra 2 . The Unit Circle Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. 9. sin 315o The angle passes through the point on the unit circle. Use sin θ = y. sin 315° = y Holt McDougal Algebra 2 The Unit Circle Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. 10. tan 180o The angle passes through the point (–1, 0) on the unit circle. Use tan θ = tan 180° = Holt McDougal Algebra 2 . The Unit Circle Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. 4 11. cos 3 The angle passes through the point on the unit circle. Use cos θ = x. 4 1 cos 3 2 Holt McDougal Algebra 2 The Unit Circle Lesson 10.3 Practice A Holt McDougal Algebra 2