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The Unit Circle

The Unit Circle. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 2. Holt Algebra 2. Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225° 3. –150° 4. 315° Find the exact value of each trigonometric function.

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The Unit Circle

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

  2. Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225° 3. –150° 4. 315° Find the exact value of each trigonometric function. 5. sin 60° 6. tan 45° 7. cos 45° 8. cos 60° 60° 45° 30° 45° 1

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

  4. Vocabulary radian unit circle

  5. 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.

  6. 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.

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

  8. 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.

  9. 4 9 . 20 . Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. a. 80° b.

  10. 5 . . Check It Out! Example 1 Convert each measure from degrees to radians or from radians to degrees. c. –36° d. 4radians

  11. 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 θin the standard position:

  12. 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.

  13. The angle passes through the point 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° cos 225° = x Use cos θ = x.

  14. The angle passes through the point on the unit circle. Use tan θ = . Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. tan

  15. The angle passes through the point on the unit circle. Check It Out! Example 1a Use the unit circle to find the exact value of each trigonometric function. sin 315° sin 315° = y Use sin θ = y.

  16. tan 180° = Use tan θ = . Check It Out! Example 1b Use the unit circle to find the exact value of each trigonometric function. tan 180° The angle passes through the point (–1, 0) on the unit circle.

  17. The angle passes through the point on the unit circle. Check It Out! Example 1c Use the unit circle to find the exact value of each trigonometric function.

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

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

  20. 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°

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

  22. 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.

  23. 270° 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°

  24. 90° Check It Out! Example 3a Continued Step 2 Find the sine, cosine, and tangent of the reference angle. sin 90° = 1 Use sin θ = y. cos 90° = 0 Use cos θ = x. tan 90° = undef.

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

  26. The reference angle measures . 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.

  27. 30° 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.

  28. Check It Out! Example 3b 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.

  29. –30° Check It Out! Example 3c 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. 30° 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.

  31. Check It Out! Example 3c 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.

  32. 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.

  33. The radius is of the diameter. 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 θthrough which the tire rotates in 1 second. Write a proportion.

  34. 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.

  35. Step 3 Find the length of the arc intercepted by radians. Substitute 0.325 for r and for θ Example 4 Continued Use the arc length formula. Simplify by using a calculator. The car travels about 22 meters in second.

  36. 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. r =14 Step 2 Find the angle θthrough which the hour hand rotates in 1 minute. Write a proportion.

  37. 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.

  38. Substitute 14 for r and for θ. 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 Simplify by using a calculator. The minute hand travels about 1.5 feet in one minute.

  39. 3. Use the unit circle to find the exact value of . Lesson Quiz: Part I Convert each measure from degrees to radians or from radians to degrees. 144° 2. 1. 100° 4. Use a reference angle to find the exact value of the sine, cosine, and tangent of

  40. Lesson Quiz: Part II 5. A carpenter is designing a curved piece of molding for the ceiling of a museum. The curve will be an arc of a circle with a radius of 3 m. The central angle will measure 120°. To the nearest tenth of a meter, what will be the length of the molding? 6.3 m

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