# what's the range of $y=\frac{\sin x+a}{\cos x+b}$ .

what's the range of $y=\frac{\sin x+a}{\cos x+b}$ .

It is a question I meet somewhere, I hope to find the most simple solution.

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What have you tried to do so far? –  Daryl Oct 19 '12 at 23:33

Hint: For the given angle $x$, we can always find numbers $a'$ and $b'$ such that $a = a'\sin x$ and $b = b'\cos x$. Given this, we may do some nice simplifications, and the solution should be clear.
Assuming $|b| > 1$, find the critical points using the derivative. The endpoints of the range will be the values of your function at the critical points.