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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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up vote 1 down vote accepted

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.

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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.

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