Finding $\arccos(\sin x) $

Question

How do I find $\arccos(\sin x) $? I have found a formula which states that $\arccos (\sin x) =\frac{\pi}{2}-x-2n \pi$ where $n\in \mathbb{Z} $, but I don't know how to choose $n$ depending on $x$'s value.

Answer 1

The answer depends upon the value of $x$. If $x\in\left[-\frac\pi2,\frac\pi2\right]$, $\arccos\bigl(\sin(x)\bigr)=\frac\pi2-x$. If $x\in\left[\frac\pi2,\frac{3\pi}2\right]$, $\arccos\bigl(\sin(x)\bigr)=x-\frac\pi2$. And so on…

And, of course, you should keep in mind that it is a periodic function with period equal to $2\pi$. So, knowing which values it takes in $\left[-\frac\pi2,\frac{3\pi}2\right]$ is enough to know the whole function.