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This question already has an answer here:

$sin(x)$ 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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marked as duplicate by anomaly, José Carlos Santos, John Omielan, Yanior Weg, Jyrki Lahtonen May 11 at 16:38

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Check out a little book called Transcendental Numbers by Siegel. $\endgroup$ – logarithm May 10 at 23:22
  • $\begingroup$ @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, $\endgroup$ – logarithm May 10 at 23:29
  • $\begingroup$ Fairly basic and inexpensive, Irrational Numbers by Niven books.google.com/books/about/… $\endgroup$ – Will Jagy May 11 at 0:21