One could use Euler's identity to show that $\sin(z)$ is an elliptic function, however, if we define the $\arcsin(z)$ as
$$\arcsin(z) = \int_0^z \frac{1}{\sqrt{1-w^2}}dw$$
we would like to show that this is a conformal map, mapping the upper half-plane to a triangle (where one of the points of the triangle is the point of infinity, so this triangle will look like an infinite parallel strip). Then from we make use of the Schwarz reflection principle, and then note that $\sin(z)$ is the inverse map.
My issue is showing why this is a conformal map from the upper half-plane to the triangle.