Diophantine equation: $|\sin a|=|\sin b|^c$

Question

Does there exist integer solutions to $$|\sin a|=|\sin b|^c$$ other than $a=b$, $c=1$?

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Currently I have no progress. To merely satisfy the requirements of MSE, I can only say that I invented this problem when I try to create Diophantine equations that involve special functions.

I apologize for that.

Thanks for any help in advance.

Answer 1

The only other solutions would be $a=b=0$ and $c \in \mathbb{Z}$.

Answer 2

For the diophantine equation $$ |\sin(a)|=|\sin(b)|^c $$ there are some obvious classes of solutions:

1) Take $a=b$ and $c=1$, as in the OP.

2) Take $a=-b$ and $c=1$.

3) Take $a=b=0$ and $c\in\mathbb{Z}\setminus\{0\}$ as in another answer.

What else can we say? We know that $\sin(x)$ is transcendental at non-zero integer values, so $c=0$ can have no solutions. I will leave $0^0$ undefined, but if you do define it you might get an extra solution $(0=0^0)$.

Can there be any other solutions for $c\neq0,1$? Clearly $a\neq b$ is required, but beyond this (if I recall my transcendental number theory correctly) I think you have an open problem. We do not know that the values of $\sin(x)$ at different integers are algebraically independent, so there might be a solution or there might not be.