B Apply

Determine the number of cycles each sine function has in the interval from 0 to 2π. Find the amplitude and period of each function.

31. y = sin θ y equals sine theta
32. y = sin 5 θ y equals sine 5 theta
33. y = sin π θ y equals sine pi theta
34. y = 3 sin θ y equals 3 sine theta
35. y = − 5 sin θ y equals negative 5 sine theta
36. y = − 5 sin 2 π θ y equals negative 5 sine 2 pi theta
37. Graphing Calculator Graph the functions y = 3 sin θ y equals 3 sine theta  and y = − 3 sin θ y equals negative 3 sine theta  on the same screen. How are the two graphs related? How does the graph of y = a sin b θ y equals eh sine b theta  change when a is replaced with its opposite?
38. Use the formula period = 2 π b period , equals , fraction 2 pi , over b end fraction  to find the period of each sine function.
    1. y = 1 . 5 sin 2 θ y equals 1 . 5 sine 2 theta
    2. y = 3 sin π 2 θ y equals 3 sine , pi over 2 , theta
39. Think About a Plan The sound wave for the note A above middle C can be modeled by the function y = y equals  0.001 sin 880πθ. Sketch a graph of the sine curve.
    • What is the period of the function?
    • What is the amplitude of the function?
    • How many cycles of the graph are between 0 and 2π?

C Challenge

Find the period and amplitude of each sine function. Then sketch each function from 0 to 2π.

40. y = − 3 . 5 sin 5 θ y equals negative 3 . 5 sine 5 theta
41. y = 5 2 sin 2 θ y equals , 5 halves sine 2 theta
42. y = − 2 sin 2 π θ y equals negative 2 sine 2 pi theta
43. y = 0 . 4 sin 3 θ y equals 0 . 4 sine 3 theta
44. y = 0.5 sin π 3 θ y equals 0.5 sine , pi over 3 , theta
45. y = − 1.2 sin 5 π 6 θ y equals negative 1.2 sine . fraction 5 pi , over 6 end fraction , theta
46. Open-Ended Write the equations of three sine functions with the same amplitude that have periods of 2, 3, and 4. Then sketch all three graphs.
47. Music The sound wave for a certain pitch fork can be modeled by the function y = 0 . 001 sin 1320 π θ . y equals 0 . 001 sine , 1320 , pi theta .  Sketch a graph of the sine curve.

48. Astronomy In Houston, Texas, at the spring equinox (March 21), there are 12 hours and 9 minutes of sunlight. The longest and shortest day of the year vary from the equinox by 1 h 55 min. The amount of sunlight during the year can be modeled by a sine function.

    

    1. Define the independent and dependent variables for a function that models the variation in hours of sunlight in Houston.
    2. What are the amplitude and period of the function measured in days?
    3. Write a function that relates the number of days away from the spring equinox to the variation in hours of sunlight in Houston.
    4. Estimation Use your function from part (c). In Houston, about how much less sunlight does February 14 have than March 21?

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