Question

How do you evaluate `arcsin 0`?

Answer 1

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

1) `arcsin(x)={alpha in RR : sin(alpha)=x}`
In this case `arcsin(0)=piZZ={x in RR :x=pik, kinZZ}` (if you have never seen the `piZZ` notation just ignore it)

2) `arcsin(x)=alpha` (the only `alpha` in `[-pi/2,pi/2]` such that `sin(alpha)=x`)
In this case `arcsin(0)=0`

N.B.
I usually write 2) as Arcsin

Notice that Arcsin(x)`in arcsin(x) forall x`