Graph of $\arcsin(\cos(x))$

Question

How to draw graph of $\arcsin(\cos(x))$ or even $\arcsin(\sin(x))$ without use of graphing calculator , its sort of confusing me from long time. It gives pointed curves when drawing from calculator , why does it looks like that? Why it isn't just linear graph? Can someone shed some light here?

Answer 1

Keep in mind that the domain of the arcsin function is $[-1,1]$ and its range is $[-\pi/2,\pi/2]$. This is important because even though $\sin x$ and $\arcsin x$ are inverse functions, it's not correct to say that $\arcsin(\sin x)=x$ for all $x$.

This might explain why your graphing calculator is giving you "pointed" curves.

Answer 2

Hint:
For the first function use the identity $$\cos(x)=\sin\left(x+\dfrac\pi2\right),$$ for the second it is useful to note that $\arcsin(\sin(x))=x$, since $\arcsin x$ and $\sin x$ are inverse functions.

Also be careful about the domains and ranges of those functions, and so $\arcsin(\sin(x))=x$ isn't true for all $x$.