Consider a spherical triangle with 3 angles: $\pi/2$, $\pi/3$, $\pi/3$. All sides are geodesics of course. The sphere has radius $r=1$.
Context: I want to know whether a given point in polar coordinates ($\phi$, $\theta$, $r$) where $r=1$, is in my triangle. (My convention for polar coordinates is: $\phi$ is the rotational angle around the north pole and $\theta$ is the azimuthal angle. $\theta=0$ is the north pole. $\theta=\pi$ is the south pole).
I will place the right angle at the north pole. One side of my triangle will be at $0$ longitude, and one will be at $\pi/2$ longitude. So clearly one condition must be: $0<\phi<\pi/2$.
But I can't figure out the condition that determines whether I am north of my geodesic that defines the base/hypotenuse of the triangle. The condition would be in terms of theta which would in turn be a function of $\phi$, but that's as far as I can get.
Thank you in advance for any help you can provide.