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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    $\begingroup$ The $\sin 72^\circ$ 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 $\sin$. $\endgroup$ – Noble Mushtak Jun 13 '16 at 3:25
  • $\begingroup$ i am sorry i meant to calculate sin24 to see 72 can be trisected. i editted the question $\endgroup$ – Mathcho Jun 13 '16 at 3:28
  • $\begingroup$ A pentadecagon can be constructed. $\endgroup$ – Joffan Jun 13 '16 at 3:33
  • $\begingroup$ 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. $\endgroup$ – Blue Jun 13 '16 at 3:48
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    $\begingroup$ $$\sin(60-36)^\circ=?$$ and use intmath.com/blog/mathematics/… $\endgroup$ – lab bhattacharjee Jun 13 '16 at 5:14

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.

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