How can I prove that $\sin (10^\circ), \sin(1^\circ), \sin(2^\circ), \sin(3^\circ), \tan(10^\circ)$ are irrational

Prove that $\sin (10^\circ)$, $\sin(1^\circ)$, $\sin(2^\circ)$, $\sin(3^\circ)$, and $\tan(10^\circ)$ are irrational.

My Attempt:

Let $x = 10^\circ$. Then

\begin{align} x &= 10^\circ\\ 3x &= 30^\circ\\ \sin (3x) &= \sin (30^\circ)\\ 3\sin (10^\circ)-4\sin^3(0^\circ) &= \frac{1}{2} \end{align}

Now let $y = \sin (10^\circ)$. Then

\begin{align} 3y-4y^3 &= \frac{1}{2}\\ 6y-8y^3 &= 1\\ \tag18y^3-6y+1 &= 0 \end{align}

How can I calculate the roots of $(1)$?

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You don't need to find the roots, you just need to show that they aren't rational. – Git Gud May 19 '13 at 16:00
I'm assuming you mean, e.g. $\sin(10^\circ)$ and not $\sin(10^0)$? – amWhy May 19 '13 at 16:01
Thanks Git Gud but how can i prove that they are not rational – juantheron May 19 '13 at 16:02

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