I am interested in finding exact values for the angles of a 3-4-5 triangle.
In particular, I would like to know the exact value of $\frac{1}{4}\sin^{-1}(\frac{4}{5})+\sin^{-1}(\frac{3}{5})$.
For context, this came up in an integral i was solving, mainly for fun. Here is the integral, in case there is a simpler solution:
$$\frac{1}{2}\int_0^{\frac{2}{5}}1+f(2t)dt+\int_{\frac{2}{5}}^{\frac{1}{2}}f(t-1)dt-\frac{1}{2}\int_0^{\frac{1}{2}}f(2t-1)dt$$
where $f(x)=\sqrt{1-x^2}$.
I've looked at this question: Prove that the ratio of acute angles in a $3:4:5$ triangle is irrational, so I understand if what I'm asking for is not possible.