13-3 The Unit Circle Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

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

Vocabulary radian unit circle

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.

The circumference of a circle of radius r is 2r 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.

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

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

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

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:

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.

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.

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 = .

Check It Out! Example 1a Use the unit circle to find the exact value of each trigonometric function. sin 315°

Check It Out! Example 1b Use the unit circle to find the exact value of each trigonometric function. tan 180°

Check It Out! Example 1c Use the unit circle to find the exact value of each trigonometric function.

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

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

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°

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

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.

Check It Out! Example 3a Use a reference angle to find the exact value of the sine, cosine, and tangent of 270°.

Check It Out! Example 3b Use a reference angle to find the exact value of the sine, cosine, and tangent of each 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°

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.

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.

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.

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.

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?