TL;DR: Are there shorter, established notations for the sine function and other trigonometric functions than "$ \sin(x) $" etc.?
In a some homeworks and tests, there are exercises (e.g. differential equations or matrices) which contains sine and cosine, which I have to write very often - obviously, this becomes quite annoying.
My usual "workaround" is, that I define $ \sin(x) = s $ and $ \cos(x) = c $ to safe me from writing the rest and loose time unessecarily. As far as I can see, this is totally correct and can't be marked as wrong.
Yet this reduces the readability (and probably annoys the corrector). So are there any shorter notations for the trigonometric functions, which are established or at least sometimes used?
In general, there is no such other notation. Your relabelling approach is certainly fine (except maybe if the work is being marked by someone unwilling to look through your custom symbol definitions). But I'd concentrate on whether your proofs are more verbose than necessary for other reasons. Writing in terms of $e^{ix}$ would shorten many proofs, especially if it's appropriate in that context to write e.g. $z=e^{ix}$.
In contexts where multiple angles are indexed viz. $\theta_i$, it's common to use notation such as $c_i:=\cos\theta_i$ etc. See e.g. the tabulated results in terms of proper Euler angles & Tait–Bryan angles here. Credit goes to @JohnDouma for this observation.