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

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

My try:: Let $x = 10^\circ$, Then $3x = 30^\circ$

Now $\sin (3x) = \sin (30^\circ)$

So $\displaystyle 3\sin (10^\circ)-4\sin^3(0^\circ) = \frac{1}{2}$

Let $x = \sin (10^\circ)$

Then $\displaystyle 3x-4x^3 = \frac{1}{2}\Leftrightarrow 6x-8x^3=1$

$8x^3-6x+1 = 0$

Now How Can i calculate Roots of Given equation.

plz explain me

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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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