Is there any method to do it by hand quickly? i want to show the angle $72$ can be trisected by compass and ruler. so i need to find the way to calculate it... help please!
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1The $\sin 72^\circ$ is much easier to find than $\sin 24^\circ$. This answer shows that $\cos\frac{\pi}{10}=\cos 18^\circ=\sin 72^\circ=\sqrt{\frac{5+\sqrt{5}}{8}}$, confirming that $72^\circ$ is constructible since it only has square roots in its $\sin$. – Noble Mushtak Jun 13 '16 at 03:25
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i am sorry i meant to calculate sin24 to see 72 can be trisected. i editted the question – Mathcho Jun 13 '16 at 03:28
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A pentadecagon can be constructed. – Joffan Jun 13 '16 at 03:33
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Related: This answer of mine shows a general form of values of $\sin k 3^\circ$ for integer $k$. Since the values involve nothing more complicated than square roots, they're constructible. – Blue Jun 13 '16 at 03:48
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2$$\sin(60-36)^\circ=?$$ and use http://www.intmath.com/blog/mathematics/how-do-you-find-exact-values-for-the-sine-of-all-angles-6212 – lab bhattacharjee Jun 13 '16 at 05:14
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We know that $60^\circ$ can be constructed and that $72^\circ$ can be constructed. By bisecting the $72^\circ$, we get $36^\circ$ and by subtracting the $60^\circ$ by the $36^\circ$, we get $24^\circ$. Thus, $24^\circ$ is constructable and $72^\circ$ can be trisected.
Noble Mushtak
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