Question

How do you evaluate `arcsin(0)` without a calculator?

Answer 1

`arcsin(0) = 0,pi,2pi,3pi...`

Explanation:

`arcsin` is the inverse of `sin`, so I prefer calling it `sin^-1`, which is what I'll use.

If we say that the solution (which we don't know yet) is `x`, then

`sin^-1(0)=x`

so, taking the `sin` of both sides,

`sin(sin^-1(0)) = 0 = sin(x)`

Effectively, we are finding the point where `sin(x)=0`. You can do this if you know what a `sin` graph looks like already, or by using a `sin` graph.

graph{sin(x) [-10, 10, -5, 5]}

Using this graph, we can see that the curve `sin(x)` is `0` at the points `(0,0)`, `(pi,0)`, `(2pi,0)`, etc.

Therefore,

`sin^-1(0)=0, pi, 2pi, 3pi, 4pi...`

or

`sin^-1(x) = npi`, where `n` is any integer (positive or negative, as the graph extends in both directions).