What is function $\sin$ without $x$? I'm solving a problem set in ODE:



This is the first time in my life that I see such functions $\sin,\cos,\tan$ without argument $x$.

I would like to ask if $\sin,\cos,\tan$ mean $\sin x,\cos x,\tan x$ or they mean something else.

Thank you so much!
Usually the convention is that if $f:X\to Y$ is a function, then $f(x)\in Y$ is the value at the point $x\in Y$. As you can do arithmetic with function values, you can do arithmetic with functions, applying the operation pointwise. Thus $h=f+g$ is $h(x)=f(x)+g(x)$ for all $x$.

This gives a little complication if you want to write equations like 1) without too much decorations like writing $\forall x\in X$. One possible interpretation would then be that $x$ has a double meaning, it is the variable name and, where appropriate, a function itself, the identity function. So that 1) would be in this convention $xy'+y=2x$, avoiding the function evaluation notation.

In 3-5, the pendulum is moving to the extreme other side regarding the function evaluation notation in that $\sin$ etc. is used as function name in the same way as $f$ above. This is a very unusual notation, but not formally wrong. But it will look very strange in larger terms, $\sin+\cos$ is tolerable, but how to interpret $\sin\cos$?
$\sin$ is an abbreviation for $\sin(x)$, in the same manner that $y'$ is an abbreviation for $y'(x)$ and $y$ for $y(x)$.