TITLE: Are all sine functions odd?
QUESTION [7 upvotes]: If I have a function like : $f(x) = \sin(e^{x/2} + e^{-x/2})$ or something equally complicated, do I actually need to work out if $f(-x) = -f(x)$, or are all sine functions odd no matter what it is a function of and it is just a matter of proving this using trig identities?
thanks!

REPLY [12 votes]: No, not all $\sin(f(x))$ are odd. In fact, you need $f$ to be odd for that to happen.
Well, not exactly; the non-injectiveness of the sine function means there are other ways to make it happen. For instance, we can loosen the oddness requirement on $f$ to "for any $x$ there is an $n$ such that $f(-x)=2\pi n-f(x)$". And that's not the most loose construction. But we're getting close.
So if you insert, say, an even $f$ into the sine function, the result is necessarily an even function. For instance, $\sin((-x)^2)=\sin(x^2)$.