13 3 The 13 3 The Unit Circle

13 -3 The 13 -3 The. Unit. Circle Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

13 -3 The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Holt Algebra 2

13 -3 The Unit Circle Vocabulary radian unit circle Holt Algebra 2

13 -3 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, then the measure of θ is defined as 1 radian. Holt Algebra 2

13 -3 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 Algebra 2

13 -3 The Unit Circle Holt Algebra 2

13 -3 The Unit Circle Example 1: Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. A. – 60° . B. Holt Algebra 2

13 -3 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 Algebra 2

13 -3 The Unit Circle Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. a. 80° b. c. – 36° d. 4 radians Holt Algebra 2

13 -3 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: Holt Algebra 2

13 -3 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 Algebra 2

13 -3 The Unit Circle Example 2 A: 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 Holt Algebra 2 Use cos θ = x.

13 -3 The Unit Circle Example 2 B: 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 = Holt Algebra 2 .

13 -3 The Unit Circle Check It Out! Example 1 a Use the unit circle to find the exact value of each trigonometric function. sin 315° Holt Algebra 2

13 -3 The Unit Circle Check It Out! Example 1 b Use the unit circle to find the exact value of each trigonometric function. tan 180° Holt Algebra 2

13 -3 The Unit Circle Check It Out! Example 1 c Use the unit circle to find the exact value of each trigonometric function. Holt Algebra 2

13 -3 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 Holt Algebra 2

13 -3 The Unit Circle The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position. Holt Algebra 2

13 -3 The Unit Circle 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° Holt Algebra 2

13 -3 The Unit Circle Example 3 Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. Holt Algebra 2

13 -3 The Unit Circle 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. Holt Algebra 2

13 -3 The Unit Circle Check It Out! Example 3 a Use a reference angle to find the exact value of the sine, cosine, and tangent of 270°. Holt Algebra 2

13 -3 The Unit Circle Check It Out! Example 3 b Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Holt Algebra 2

13 -3 The Unit Circle Check It Out! Example 3 c Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. – 30° Holt Algebra 2

13 -3 The Unit Circle 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. Holt Algebra 2

13 -3 The Unit Circle Holt Algebra 2

13 -3 The Unit Circle 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 diameter. of the Step 2 Find the angle θ through which the tire rotates in 1 second. Write a proportion. Holt Algebra 2

13 -3 The Unit Circle 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. Holt Algebra 2

13 -3 The Unit Circle Example 4 Continued Step 3 Find the length of the arc intercepted by radians. Use the arc length formula. Substitute 0. 325 for r and for θ Simplify by using a calculator. The car travels about 22 meters in second. Holt Algebra 2

13 -3 The Unit Circle 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? Holt Algebra 2