UNIT CIRCLE Continue Objectives Convert angle measures between

UNIT CIRCLE (Continue)

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

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 3 a Use a reference angle to find the exact value of the sine, cosine, and tangent of 270°. Step 1 Find the measure of the reference angle. The reference angle measures 90° 270°

Check It Out! Example 3 a Continued Step 2 Find the sine, cosine, and tangent of the reference angle. sin 90° = 1 Use sin θ = y. 90° cos 90° = 0 tan 90° = undef. Use cos θ = x.

Check It Out! Example 3 a Continued Step 3 Adjust the signs, if needed. sin 270° = – 1 cos 270° = 0 tan 270° = undef. In Quadrant IV, sin θ is negative.

Check It Out! Example 3 b Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Step 1 Find the measure of the reference angle. The reference angle measures.

Check It Out! Example 3 b Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. 30°

Check It Out! Example 3 b 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 3 c Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. – 30° Step 1 Find the measure of the reference angle. The reference angle measures 30°. – 30°

Check It Out! Example 3 c Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. 30°

Check It Out! Example 3 c 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.

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 diameter. of the 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 A 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? Step 1 Find the radius of the clock. The radius is the actual r =14 length of the hour hand. Step 2 Find the angle θ through which the hour hand rotates in 1 minute. Write a proportion.

Check It Out! Example 4 Continued The hand rotates θ radians in 1 m and 2 radians in 60 m. Cross multiply. Divide both sides by 60. Simplify.

Check It Out! Example 4 Continued Step 3 Find the length of the arc intercepted by radians. Use the arc length formula. s ≈ 1. 5 feet Substitute 14 for r and for θ. Simplify by using a calculator. The minute hand travels about 1. 5 feet in one minute.