Practice and Problem-Solving Exercises

A Practice

Use a unit circle, a 30 ° − 60 ° − 90 ° 30 degrees negative 60 degrees negative 90 degrees triangle, and an inverse function to find the degree measure of each angle. See Problem 1.

7. angle whose sine is 1
8. angle whose tangent is 3 3 fraction square root of 3 , over 3 end fraction
9. angle whose sine is − 3 2 negative , fraction square root of 3 , over 2 end fraction
10. angle whose tangent is − 3 negative square root of 3
11. angle whose cosine is 0
12. angle whose cosine is − 2 2 negative , fraction square root of 2 , over 2 end fraction

Use a calculator and inverse functions to find the radian measures of all angles having the given trigonometric values. See Problems 2 and 3.

13. angles whose tangent is 1
14. angles whose sine is 0.37
15. angles whose sine is − 0.78 negative , 0.78
16. angles whose tangent is − 3 negative 3
17. angles whose cosine is − 0.89 negative , 0.89
18. angles whose sine is − 1.1 negative 1.1

Solve each equation for θ theta with 0 ≤ θ < 2 π . 0 less than or equal to theta less than 2 pi . See Problems 4 and 5.

19. 2 sin θ = 1 2 sine theta equals 1
20. 2 cos θ − 3 = 0 2 cosine theta negative square root of 3 equals 0
21. 4 tan θ = 3 + tan θ 4 tangent theta equals 3 plus tangent theta
22. 2 sin θ − 2 = 0 2 sine theta negative square root of 2 equals 0
23. 3 tan θ − 1 = tan θ 3 tangent theta negative 1 equals tangent theta
24. 3 tan θ + 5 = 0 3 tangent theta plus 5 equals 0
25. 2 sin θ = 3 2 sine theta equals 3
26. 2 sin θ = − 3 2 sine theta equals negative square root of 3
27. ( cos θ ) ( cos θ + 1 ) = 0 open cosine theta close open cosine theta plus 1 close equals 0
28. ( sin θ − 1 ) ( sin θ + 1 ) = 0 open sine theta negative 1 close open sine theta plus 1 close equals 0
29. 2 sin 2 θ − 1 = 0 2 , sine squared , theta negative 1 equals 0
30. tan θ = tan 2 θ tangent theta equals , tangent squared , theta
31. sin 2 θ + 3 sin θ = 0 sine squared , theta plus 3 sine theta equals 0
32. sin θ = − sin θ cos θ sine theta equals negative sine theta cosine theta
33. 2 sin 2 θ − 3 sin θ = 2 2 , sine squared , theta negative 3 sine theta equals 2
34. Energy Conservation Suppose the outside temperature in Problem 6 is modeled by the function f ( t ) = 27 − 6 cos π 12 f open t close equals 27 minus 6 . cosine , fraction pi , over 12 end fraction instead. During what hours is the air conditioner cooling the house? See Problem 6.

B Apply

Each diagram shows one solution to the equation below it. Find the complete solution of each equation.

35. 

    5 sin θ = 1 + 3 sin θ 5 sine theta equals 1 plus 3 sine theta

36. 

    6 cos θ − 5 = − 2 6 cosine theta negative 5 equals negative 2

37. 

    4 sin θ + 3 = 1 4 sine theta plus 3 equals 1

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

38. sec θ = 2 secant theta equals 2
39. csc θ = − 1 co-secant theta equals negative 1
40. csc θ = 3 co-secant theta equals 3
41. cot θ = − 10 co-tangent theta equals negative 10

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