Use the definitions of the trigonometric ratios for a right triangle to derive a cofunction identity for each expression. See Problem 2.

13. tan ( 90 ° − A ) tangent open 90 degrees negative eh close
14. csc ( 90 ° − A ) co-secant open 90 degrees negative eh close
15. 4

Solve each trigonometric equation for θ theta with 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi . See Problem 3.

16. cos ( π 2 − θ ) = csc θ cosine . open . pi over 2 , minus theta . close . equals co-secant theta
17. sin ( π 2 − θ ) = − cos ( − θ ) sine . open . pi over 2 , minus theta . close . equals negative cosine . open , negative theta , close
18. tan ( π 2 − θ ) + tan ( − θ ) = 0 tangent . open . pi over 2 , minus theta . close . plus tangent . open , negative theta , close . equals 0
19. tan 2 θ − sec 2 θ = cos ( − θ ) tangent squared , theta negative , secant squared , theta equals cosine open negative theta close
20. 2 sin ( π 2 − θ ) = sin ( − θ ) 2 sine . open . pi over 2 , minus theta . close . equals sine . open , negative theta , close
21. tan ( π 2 − θ ) = cos ( − θ ) tangent . open . pi over 2 , minus theta . close . equals cosine . open , negative theta , close

Mental Math Find the value of each trigonometric expression. See Problems 4, 5, and 6.

22. cos 50 ° cos 40 ° − sin 50 ° sin 40 ° cosine , 50 degrees cosine , 40 degrees negative sine , 50 degrees sine , 40 degrees
23. sin 80 ° cos 35 ° − cos 80 ° sin 35 ° sine , 80 degrees cosine , 35 degrees negative cosine , 80 degrees sine , 35 degrees
24. sin 100 ° cos 170 ° + cos 100 ° sin 170 ° sine , 100 degrees cosine , 170 degrees plus cosine , 100 degrees sine , 170 degrees
25. cos 183 ° cos 93 ° + sin 183 ° sin 93 ° cosine , 183 degrees cosine , 93 degrees plus sine , 183 degrees sine , 93 degrees

Find each exact value. Use a sum or difference identity. See Problems 4, 5, and 6.

26. cos 105 ° cosine , 105 degrees
27. tan 75 ° tangent , 75 degrees
28. tan 15 ° tangent , 15 degrees
29. sin 75 ° sine , 75 degrees
30. cos 75 ° cosine , 75 degrees
31. tan ( − 15 ° ) tangent open negative 15 degrees close
32. sin 225 ° sine , 225 degrees
33. cos 240 ° cosine , 240 degrees
34. sin 390 ° sine , 390 degrees
35. cos ( − 300 ° ) cosine open negative 300 degrees close

B Apply

36. Think About a Plan At exactly 22 1 2 22 , and 1 half minutes after the hour, the minute hand of a clock is at point P, as shown in the diagram. Several minutes later, it has rotated θ theta degrees clockwise to point Q. The coordinates of point Q are ( cos − ( θ + 45 ° ) , sin − ( θ + 45 ° ) ) . open cosine minus open theta plus 45 degrees close comma sine minus open theta plus 45 degrees close close . Write the coordinates of point Q in terms of cos θ cosine theta and sin θ . sine theta .
    • What trigonometric identities can you use?
    • How can you use the diagram to check your answer?

Image Long Description

Verify each identity.

37. sin ( A − B ) = sin A cos B − cos A sin B sine open eh minus b close equals sine eh cosine b minus cosine eh sine b
38. tan ( A − B ) = tan A − tan B 1 + tan A tan B tangent . open , eh minus b , close . equals . fraction tangent eh minus tangent b , over 1 plus tangent eh tangent b end fraction
39. tan ( A + B ) = tan A + tan B 1 − tan A tan B tangent . open , eh plus b , close . equals . fraction tangent eh plus tangent b , over 1 minus tangent eh tangent b end fraction
40. sin ( x + π 2 ) + sin ( x − π 3 ) = sin x sine . open . x plus , pi over 2 . close . plus sine . open . x minus , pi over 3 . close . equals sine x

41. Gears The diagram below shows a gear whose radius is 10 cm. Point A represents a 60 ° 60 degrees counterclockwise rotation of point P(10, 0). Point B represents a θ theta -degree rotation of point A. The coordinates of B are ( 10 cos ( θ + 60 ° ) , 10 sin ( θ + 60 ° ) ) . open 10 cosine open theta plus 60 degrees close comma 10 sine open theta plus 60 degrees close close . Write these coordinates in terms of cos θ cosine theta and sin θ . sine theta .

    

Rewrite each expression as a trigonometric function of a single angle measure.

42. sin 2 θ cos θ + cos 2 θ sin θ sine 2 theta cosine theta plus cosine 2 theta sine theta
43. sin 3 θ cos 2 θ + cos 3 θ sin 2 θ sine 3 theta cosine 2 theta plus cosine 3 theta sine 2 theta
44. cos 3 θ cos 4 θ − sin 3 θ sin 4 θ cosine 3 theta cosine 4 theta negative sine 3 theta sine 4 theta
45. cos 2 θ cos 3 θ − sin 2 θ sin 3 θ cosine 2 theta cosine 3 theta negative sine 2 theta sine 3 theta
46. tan 5 θ + tan 6 θ 1 − tan 5 θ tan 6 θ fraction tangent 5 theta plus tangent 6 theta , over 1 minus tangent 5 theta tangent 6 theta end fraction
47. tan 3 θ − tan θ 1 + tan 3 θ tan θ fraction tangent 3 theta minus tangent theta , over 1 plus tangent 3 theta tangent theta end fraction

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