I want to minimize the function $a\sin (x) + b\csc(x)$ without using calculus.
I know how to minimize a function like $a\sin(x)+b\cos(x)$. I write
$$\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\sin(x)+\frac{b}{\sqrt{a^2+b^2}}\cos(x)\right) = \sqrt{a^2+b^2}\sin(x+\alpha)$$ where $\alpha=\arcsin\left(\frac{b}{\sqrt{a^2+b^2}}\right)$. So I have the minimum when $\sin(x+\alpha)=-1$.
However, I don't know how to apply a similar procedure for $a\sin (x) + b\csc(x)$. Do I have to do something similar? Or maybe I can use the AM-GM inequality? Many thanks in advance!