# How do you evaluate arcsin 0?

May 24, 2015

There are two answers for this question, they depends on the definition of $\arcsin$:

1) $\arcsin \left(x\right) = \left\{\alpha \in \mathbb{R} : \sin \left(\alpha\right) = x\right\}$
In this case $\arcsin \left(0\right) = \pi \mathbb{Z} = \left\{x \in \mathbb{R} : x = \pi k , k \in \mathbb{Z}\right\}$ (if you have never seen the $\pi \mathbb{Z}$ notation just ignore it)

2) $\arcsin \left(x\right) = \alpha$ (the only $\alpha$ in $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ such that $\sin \left(\alpha\right) = x$)
In this case $\arcsin \left(0\right) = 0$

N.B.
I usually write 2) as Arcsin

Notice that Arcsin(x)$\in \arcsin \left(x\right) \forall x$