1. Holt McDougal Algebra 2 10-3 The Unit Circle 10-3 The Unit Circle Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

2. 10-3 The Unit Circle Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Objectives

3. Holt McDougal Algebra 2 10-3 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.

4. Holt McDougal Algebra 2 10-3 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.

5. Holt McDougal Algebra 2 10-3 The Unit Circle

6. Holt McDougal Algebra 2 10-3 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..

7. Holt McDougal Algebra 2 10-3 The Unit Circle 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

8. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. a. 80° b... 4 9 20

9. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. c. – 36° d. 4 radians.. 5

10. Holt McDougal Algebra 2 10-3 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:

11. Holt McDougal Algebra 2 10-3 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 x- and y-coordinates of points on the unit circle.

12. Holt McDougal Algebra 2 10-3 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

13. Holt McDougal Algebra 2 10-3 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.

14. Holt McDougal Algebra 2 10-3 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°

15. Holt McDougal Algebra 2 10-3 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.

16. Holt McDougal Algebra 2 10-3 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.

17. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3a 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°

18. Holt McDougal Algebra 2 10-3 The Unit Circle Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. Check It Out! Example 3a Continued 90° tan 90° = undef. sin 90° = 1 cos 90° = 0

19. Holt McDougal Algebra 2 10-3 The Unit Circle Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. Check It Out! Example 3a Continued sin 270° = – 1 cos 270° = 0 tan 270° = undef.

20. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3b 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.

21. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3b Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. 30°

22. Holt McDougal Algebra 2 10-3 The Unit Circle Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. Check It Out! Example 3b Continued In Quadrant IV, cos θ is positive. In Quadrant IV, tan θ is negative.

23. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3c 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 30°. – 30°

24. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3c Continued Step 2 Find the sine, cosine, and tangent of the reference angle. Use sin θ = y. Use cos θ = x. 30°

25. Holt McDougal Algebra 2 10-3 The Unit Circle Step 3 Adjust the signs, if needed. In Quadrant IV, sin θ is negative. Check It Out! Example 3c Continued In Quadrant IV, cos θ is positive. In Quadrant IV, tan θ is negative.

26. Holt McDougal Algebra 2 10-3 The Unit Circle

27. Holt McDougal Algebra 2 10-3 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. Step 2 Find the angle θ through which the tire rotates in 1 second. The radius is of the diameter. Write a proportion.

28. Holt McDougal Algebra 2 10-3 The Unit Circle Example 4 Continued The tire rotates θ radians in 1 s and 653(2  ) radians in 60 s. Simplify. Divide both sides by 60. Cross multiply.

29. Holt McDougal Algebra 2 10-3 The Unit Circle Example 4 Continued Step 3 Find the length of the arc intercepted by radians. Use the arc length formula. Simplify by using a calculator. Substitute 0.325 for r and for θ The car travels about 22 meters in second.

30. Holt McDougal Algebra 2 10-3 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? Step 1 Find the radius of the clock. The radius is the actual length of the hour hand. Step 2 Find the angle θ through which the hour hand rotates in 1 minute. Write a proportion. r =14

31. Holt McDougal Algebra 2 10-3 The Unit Circle The hand rotates θ radians in 1 m and 2  radians in 60 m. Simplify. Divide both sides by 60. Cross multiply. Check It Out! Example 4 Continued

32. Holt McDougal Algebra 2 10-3 The Unit Circle Step 3 Find the length of the arc intercepted by radians. Use the arc length formula. Simplify by using a calculator. The minute hand travels about 1.5 feet in one minute. Check It Out! Example 4 Continued Substitute 14 for r and for θ. s ≈ 1.5 feet