It's been a while since I have done any sort of optimization problems. Consider the function:
$$f(x)=-a_1\sin(x)-a_3\sin(3x).$$
We want to find the minimum $c$ and $a_3$ at that minimum value, such that $$|-a_1\sin(x)-a_3\sin(3x)|\leq c$$ Subject to the domain constraint of $x \in [0,\alpha]$ and the constraint that $\alpha = \frac{\pi}{4}\cos^{-1}(a_1)$ and $0<\alpha< \pi/3$
I'm really not sure where to even start on this question, hopefully somebody finds this as interesting as I do!