Let $x$ be a transcendental number Algebraically Independent from $\pi$. It is known if $ \sin x $ is also transcendental or algebraic?

For example, is $\sin \sqrt{2}^\sqrt{2}\pi$ algebraic or transcendental?

NOTE: Then the sine of an transcendental number is not necessary transcendental. Are there any known example in which $\sin x$ is algebraic for another transcendental number, different of $\pi$ and that is not defined with the use of inverse trigonometric functions?


closed as off-topic by Claude Leibovici, JMP, Watson, achille hui, Morgan Rodgers May 20 '16 at 7:26

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  • $\begingroup$ @Fakemistake : maybe he meant $\frac{x}{2 \pi}$ is transcendental ? (it seems that $\sqrt{2}^{\sqrt{2}}$ is transcendental math.stackexchange.com/a/446905/276986 , https://en.wikipedia.org/wiki/Gelfond–Schneider_theorem ) $\endgroup$ – reuns May 20 '16 at 6:05
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    $\begingroup$ The are countably many algebraic and uncountably many transcendental numbers. So the statement "sin transcedental is always algebraic" cannot be true. As Fakemistake's example show, "sin transcendental is always transcedental" is also false. $\endgroup$ – achille hui May 20 '16 at 6:06
  • $\begingroup$ @achillehui : what about what I wrote ? $\endgroup$ – reuns May 20 '16 at 6:08
  • $\begingroup$ @user1952009 - I don't have an example at hand for your case. I'll wait until OP clarify what hir mean. $\endgroup$ – achille hui May 20 '16 at 6:10
  • $\begingroup$ Check out en.wikipedia.org/wiki/Schanuel%27s_conjecture $\endgroup$ – shai horowitz May 20 '16 at 7:27

$\arcsin(1/3)$ is known to be transcendental, so it is a transcendental number whose sine is algebraic, indeed, rational.


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