$f(t) = a\cos(t) + i\sin(nt).$ Determine $a$ and $n$ through a graph.

Question

I don't know how to solve this problem. I tried using Euler's formula, but that didn't seem to work out. The problem is as follows:

Given the function: $f(t)= a\cos(t)+i\sin(nt)$

where $\;i\;$ is the imaginary number and $\;a\;$ and $\;n\;$ are both positive real numbers.

Given the graph for the function, determine $\;a\;$ and $\;n\;:$

Answer 1

Two hints:

1) Look at the graph. The $a$ tells you how much the real part is stretched by, so can you tell what it is from the graph, i.e. what is the maximum possible real part?

2) It's obvious the real part goes through one period for every $2\pi$ in $t$. Graphically, trace out the curve with your finger and count, during one full period (when the real part starts and comes back to the same edge $t=0$ and $t=2\pi$) how many periods the imaginary part goes through in the same amount of time.