$sin(x)$ is the ratio between two sides of a triangle. One or more sides must be irrational for $sin(x)$ to be irrational. Why does a rational value of $x$ guarantee this?
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Check out a little book called Transcendental Numbers by Siegel. – logarithm May 10 '19 at 23:22
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@anomaly It shouldn't be marked as a duplicate of that one. The answers there only establish weaker results characterizing the rational multiples of $\pi$ that give rational outputs, but not irrationality of outputs for rational inputs, – logarithm May 10 '19 at 23:29
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Fairly basic and inexpensive, Irrational Numbers by Niven https://books.google.com/books/about/Irrational_Numbers.html?id=ov-IlIEo47cC – Will Jagy May 11 '19 at 00:21