Solve each equation in the interval from 0 to 2π. Round your answers to the nearest hundredth.

37. sin θ = 0.6 sine theta equals 0.6
38. − 3 sin 2 θ = 1.5 negative 3 sine 2 theta equals 1.5
39. sin π θ = 1 sine pi theta equals 1
40. 1. Solve − 2 sin θ = 1.2 negative 2 sine theta equals 1.2  in the interval from 0 to 2π.
    2. Solve − 2 sin θ = 1.2 negative 2 sine theta equals 1.2  in the interval 2 π ≤ θ ≤ 4 π . 2 pi less than or equal to theta less than or equal to 4 pi .  How are these solutions related to the solutions in part (a)?
41. 1. Graph the equation y = − 30 cos ( 6 π 37 t ) y equals negative 30 cosine . open . fraction 6 pi , over 37 end fraction , t . close  from Problem 3.
    2. The independent variable θ represents time (in hours). Find four times at which the water level is the highest.
    3. For how many hours during each cycle is the water level above the line y = 0 ? y equals 0 question mark  Below y = 0 ? y equals 0 question mark
42. Tides The table below shows the times for high tide and low tide of one day. The markings on the side of a local pier showed a high tide of 7 ft and a low tide of 4 ft on the previous day.

    Tide Table

    High tide	4:03 A.M.
    Low tide	10:14 A.M.
    High tide	4:25 P.M.
    Low tide	10:36 P.M.

    1. What is the average depth of water at the pier? What is the amplitude of the variation from the average depth?
    2. How long is one cycle of the tide?
    3. Write a cosine function that models the relationship between the depth of water and the time of day. Use y = 0 y equals 0  to represent the average depth of water. Use t = 0 t equals 0  to represent the time 4:03 A.M.
    4. Reasoning Suppose your boat needs at least 5 ft of water to approach or leave the pier. Between what times could you come and go?

C Challenge

43. Graph one cycle of y = cos θ , y equals cosine theta comma  one cycle of y = − cos θ , y equals negative cosine theta comma  and one cycle of y = cos ( − θ ) y equals cosine open negative theta close  on the same set of axes. Use the unit circle to explain any relationships you see among these graphs.

44. Biology A helix is a three-dimensional spiral. The coiled strands of DNA and the edges of twisted crepe paper are examples of helixes. In the diagram, the y-coordinate of each edge illustrates a cosine function. Write an equation for the y-coordinate of one edge.

    

45. 

    1. Graphing Calculator Graph y = cos θ y equals cosine theta  and y = cos ( θ − π 2 ) y equals cosine . open . theta minus , pi over 2 . close  in the interval from 0 to 2π. What translation of the graph of y = cos θ y equals cosine theta  produces the graph of y = cos ( θ − π 2 ) ? y equals cosine . open . theta minus , pi over 2 . close . question mark
    2. Graph y = cos ( θ − π 2 ) y equals cosine . open . theta minus , pi over 2 . close  and y = sin θ y equals sine theta  in the interval from 0 to 2π. What do you notice?
    3. Reasoning Explain how you could rewrite a sine function as a cosine function.

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