Is this true about the inverse sine?

Question

It is known that $ \sin(-x)=-\sin x \ $. Bbut when we say:

$$ \arcsin(-x)=-\arcsin x$$

Is this true? Is it the same with the other trigonometric functions "inverse"?

Answer 1

The inverse of any odd function (if it exists) is also an odd function.

If $f(x)$ is odd and $y = f(x)$ it must be that $-y = f(-x)$.

Using the inverse function we can say that $x = f^{-1}(y)$ and that $-x = -f^{-1}(y)$. Now take the inverse of the second equation and find that $f^{-1}(-y) = f^{-1}(f(-x)) = -x$.

We have now shown that $f^{-1}(y) = x$ and $f^{-1}(-y) = -x$ so the inverse must be an odd function.

Answer 2

Let $\theta=\arcsin{(-x)}$. Then $$\sin\theta=-x$$ $$-\sin\theta=x$$ $$\sin(-\theta)=x$$ $$-\theta=\arcsin{x}$$ $$\theta=-\arcsin{x}$$ Therefore, $-\arcsin{x}=\arcsin{(-x)}$.