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I just thought about the following expression:

$\forall x, y \in\mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y)$

I think it is true because values of $\sin(x)$ only repeat every $\pi\times n$th time, which is never reached by any rational number.

Is this true? Or am i wrong?

Thank you

Edward Jiang
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Kevin Steiger
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  • Your argument would apply equally to $\cos(x)$, but your conclusion would be false (because $\cos(x) = \cos(-x)$, and if $x$ is rational, then so is $-x$). So you have to be a bit more careful. – TonyK Oct 13 '14 at 22:02

1 Answers1

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Note that

$$\sin x - \sin y = 2 \sin \tfrac{x-y}{2}\cos \tfrac{x+y}{2}.$$

A product is $0$ if and only if at least one factor is $0$. What are the zeros of $\sin$ and $\cos$? Which are rational?

Daniel Fischer
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  • Thanks, this formula helped me:) The only rational root of $sin(x)$ is $0$ and $cos(x)$ has none, so the expression is only zero if $x=y$. – Kevin Steiger Oct 14 '14 at 21:25