The problem asks me to find a conformal mapping from the intersection of 2 half-planes: {y<2x and y>-2x} (actually it's just a triangle shape plane on the right of the y-axis) onto the right half-plane {x>0}.
I am not sure how to, perhaps, connect this problem with a triangle to half-plane question. I mean, it seems not hard to find a map from this plane to its finite subset, a similar shape triangle by letting all x larger than a certain value be such certain constant value, hence it will become a triangle. However, by doing so, will that make it a non-conformal mapping?
Choose a branch of the logarithm on the slit plane $\mathbb C- \{ z\leq 0\}$ say $\log (re^{i\theta})=\log r+i \theta $ where $\theta \in (-\pi,\pi)$
Your domain is $D:=\{ z : \Re \ z >0, \ \arg z\in (-\theta_0,\theta_0) \}$ where $\theta_0 \in (0,\frac {\pi}{2}) $
Then $\log D$ is the domain $\{z \in \mathbb C: \Im \ z \in (-\theta_0, \theta _0) \} $
Now take a dilation $z \mapsto \lambda z $ so that the image is $\{z \in \mathbb C: \Im \ z \in (-\frac{\pi}{2}, \frac{\pi}{2}) \} $
Apply exponential map. This gives you the right half plane.