What is the equation of the Cosine function that describes the frequency of the siren?

Question

A person who was listening to a siren noticed that the frequency fluctuated with time, measured in seconds. The minimum frequency was 500 Hz and the maximum frequency was 1000 Hz. The maximum frequency occurred at t=0 and again every 3 seconds. What is the equation of the cosine function that describes the frequency of this siren?

Answer 1

The generic cosine function is f(x) = Acos(B(x - C)) + D where:

A is the amplitude

B is 2π/period

C is the horizontal shift

D is the vertical shift (midline location)

In your problem, we can calculate D = 750 Hz since that is halfway between the max and min. We can calculate that A = 250 Hz since the max or min is 250 from the midline. We can deduce that C = 0 since the function maximum is at t = 0 and that's what the basic cosine function does. We can calculate B as 2π/3 since the period is 3 seconds. Then we can put these all together:

f(t) = 250cos(2π/3t) + 750