A sinusoidal function has a maximum at the point ( π/4 , 3) and the next minimum at the point ( 7π/12 , −1). Determine a cosine equation for this function.

Question

please help :)

Answer 1

For a cosine expression:

y = A cos(B(x+C)) + D

Where A is the amplitude, 2π/B is the period, C is the phase shift left and D is the vertical shift up.

We know cos(θ) normal max and min is 1 and -1. For the points given, there is a difference of 4 between the maximum and minimum points, so we need to scale cosine by an amplitude of 4/2 = 2 and then vertically shift it up by one.

So far we have y = 2cos(θ)+1. We just need to find a value for θ.

We know that half a period, max to min, is 7π/12 - π/4 = π/3 = (1/2)2π/B, so B = 3.

y = 2cos(3(x + C)) + 1

cos(0) = 1, so we want 3(π/4) + 3C = 0 --> C = -π/4

Finally, y = 2cos(3(x-π/4)) + 1

Checking:

y(π/4) = 2cos(0)+1 = 2(1)+1 = 3

y(7π/12) = 2cos(3(7π/12 - 3π/12)) + 1 = 2cos(3(π/3)) + 1 = 2cos(π) + 1 = -2 + 1 = -1