Exact angles of a 3-4-5 triangle

Question

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.

See page 4 of this for the taylor series of the inverse sine function (e.g. arc$\sin(3/5)$. — user2661923, Dec 04 '20 at 00:27
@user2661923 I know how to get an approximate vale. The question is about getting an exact solution. — Sebastián Mestre, Dec 04 '20 at 00:49
Isn't the attempt to get an exact value of arc$\sin(3/5)$ analagous to attempting to get an exact value of $e$ or $\pi$? — user2661923, Dec 04 '20 at 02:53
As noted in this answer to the referenced $3$-$4$-$5$ triangle question, we know from Niven's Theorem that $p:=\sin^{-1}(4/5)$ and $q:=\sin^{-1}(3/5)$ are not rational-degree angles (aka, rational multiples of $\pi$). Moreover, considering OP's ultimate target value $r:=\frac14p+q$, we can calculate $\sin(4r)=-44/125$; again by Niven, $4r$ is not a rational-degree angle, so that neither is $r$. — Blue, Dec 04 '20 at 06:03
@user2661923 an answer in terms of pi and e would be welcome, too. — Sebastián Mestre, Dec 04 '20 at 19:51
@Blue I see, thank you. If you post that as an answer I will mark it as accepted — Sebastián Mestre, Dec 04 '20 at 19:59

score 2 · Accepted Answer · answered Dec 04 '20 at 20:13

As noted in this answer to the question "Prove that the ratio of acute angles in a $3:4:5$ triangle is irrational", we know from Niven's Theorem that $p:=\sin^{−1}(4/5)$ and $q:=\sin^{−1}(3/5)$ are not rational-degree angles (aka, rational multiples of $\pi$), because their sines are rationals other than $0$, $\pm1/2$, and $\pm 1$.

Moreover, considering OP's ultimate target value $r:=\frac14p+q$, we can calculate $\sin(4r)=−44/125$. Again by Niven, we conclude that $4r$ is not a rational-degree angle, so that neither is $r$. $\square$

Exact angles of a 3-4-5 triangle

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