Range of function $a \sin(mx) + b \cos(nx)$ https://ift.tt/eA8V8J

What's the range of function $a \sin(mx) + b \cos(nx)$ where $a,b,m,n \in R$?

Not hard to solve for the case where $m=n$. We can let $m=n=1$ WLOG

$a \sin(x) + b \cos(x) = \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}} \sin(x) + \frac{b}{\sqrt{a^2 + b^2}} \cos(x) \right)$

Then we can substitute $\sin\theta = \frac{b}{\sqrt{a^2 + b^2}}$

How do we handle the case when they are different?

from Hot Weekly Questions - Mathematics Stack Exchange
R.Yeh