I am using spherical coordinate system $r,\theta, \phi$ like this picture.enter image description here

Consider I have a sphere with radius $R$, is divided into two hemisphere $S$ and $P$.enter image description here

With this given information I am able to calculate $\phi$ limits for $P$ portion over $T$ using the following method:

$$ \begin{split} \tan \beta = z/y,\\ z=R\cos\theta, \end{split}\quad \iff \quad y={R\cos\theta\over\tan\beta} $$

$$ \begin{split} R\sin\theta\sin\phi={R\cos\theta\over\tan\beta}, \\ \sin\phi={1\over\tan\beta\tan\theta}, \end{split}\quad \iff \quad \phi=\arcsin{1\over\tan\beta\tan\theta}.$$ So the $\phi$ can go until $\pi -\phi$.

Now consider my $P$ is not a hemisphere: rather it is a spherical cap, where the cap size can be defined by an angle $\gamma$.

enter image description here

What will be my $\phi$ in this case to calculate the $P$ portion over the line $T$?