Question

Explain why arcsin(sin(3π/4)) ≠ 3π/4 when y = sin(x) and y = arcsin(x) are inverses.

Answer 1

Short answer: $\sin x$ and $\arcsin x$ are *not* inverses. At least not perfect ones.

$\sin x$ is a multi-valued function, i.e. not one-to-one, so it's not invertible.

On the other hand, you can restrict the domain of $\sin x$ to a subset of the standard domain (all real numbers) over which $\sin x$ is not multi-valued. The standard choice for this is to consider $-\dfrac\pi2\le x\le\dfrac\pi2$. If you have a calculator with the $\arcsin$ function built into it, this is the definition it uses. Notice that $x=\dfrac{3\pi}4$ is outside this restricted domain.

This is why

$\arcsin\left(\sin\dfrac{3\pi}4\right)=\arcsin\left(\dfrac1{\sqrt2}\right)=\dfrac\pi4\neq\dfrac{3\pi}4$