I have the following problem. There is a unit hemisphere cut by the plane passing through the diameter.
The angle $\gamma$ is given. The plane cuts a half of the great circle. I need to find the area $S$ of the surface enclosed by the thick black line. Obvioulsy it is sufficient to find the area $S_1$ in one of the quadrants. To do this I define the curve $P_1P_2$ as $\theta=f(\phi)$ and the following integral has to be calculated $$S_1=\int_{0}^{\pi/2}d\phi\int_{0}^{f(\phi)}\sin\theta d\theta.$$
The problem is I don't know how to find the $f(\phi)$ function. I have tried $$\theta(\phi)=\arctan(\frac{\tan\gamma}{\sin\phi}),$$ but it did not work.