Which value is correct for $\sin^{-1}\left(\frac{-\sqrt 3}2\right)$, $\frac{4π}3$ or $\frac{5π}3$?

Question

As far as I remember, both $\frac{4π}3$ and $\frac{5π}3$ are correct for the exact values of $\sin^-1(\frac{-\sqrt 3}2)$. In symbolab and wolfram, they both showed $\frac{-π}3$, which is equivalent to $\frac{5π}3$. But in other websites, it's $\frac{4π}3$. I really got confused. I remember my teacher in Precalculus, she said it's $\frac{4π}3$. I am now confused. Which is correct?

Answer 1

$\sin$ is a function from $\mathbb R$ onto $[-1,1]$. It is not one-to-one. In order to define its inverse we have to restrict its domain. Unfortunately, not everyone uses the same domain. If you take the domain as $[-\pi /2,\pi/2]$ then the answer is $-\frac {\pi} 3$.