Why is it that when we take derivatives of trig functions we also have to take the derivative of its angle? It is because we want to know how also the angle changes?
I believe you are referring to the "chain rule". That does not apply just to trig functions, but to any functions: if x is a function of t, x(t), then $\displaystyle \frac{df(x)}{dt}= \frac{df}{dx}\frac{dx}{dt}$.
In particular, if the variable is an "angle" (actually, most applications of trig functions do not involve angles), $\displaystyle \theta(t)$ that depends on t, the the derivative of $\displaystyle sin(\theta(t))$, with respect to t, is
$\displaystyle \frac{dsin(\theta)}{dt}= \frac{dsin(\theta)}{d\theta}\frac{d\theta}{dt}= cos(\theta)\frac{d\theta}{dt}$.