SOLUTION: A sinusoidal function has an amplitude of 3 units, a maximum at (0,4), and a period of 180 degrees. Represent the function using both sine and cosine functions.

Question 1116894: A sinusoidal function has an amplitude of 3 units, a maximum at (0,4), and a period of 180 degrees. Represent the function using both sine and cosine functions.
Found 2 solutions by josmiceli, greenestamps:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

We want equations in the form

and

The function has an amplitude of 3; so a in both equations is 3.

The function has a maximum value of 4. An amplitude of 3 and a maximum value of 4 means the minimum value is -2, and the centerline (d in each equation) 1.

The period is 180 degrees, which is half the period of the basic sine or cosine function. That means b is 2.

Those are the relatively easy parts of the problem. We have as the two equations

and

By far the hardest part (for most students) is finding the values for c in each equation. That value determines the phase (horizontal) shift for the function.

The given function has a maximum at x=0. The basic cosine function has a maximum value at x=0, so there is no phase shift. So the cosine equation for your example is

For the sine function, the maximum occurs 1/4 of the way through the period; since the period of the function is 180 degrees (pi radians), the phase shift is 45 degrees (pi/4 radians) to the left. So the phase shift c is -pi/4 (making "x-c" equal to "x+pi/4"), and the sine function is

or

The first form is more meaningful to me, because I can see the phase shift of pi/4 clearly.
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