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# Famous five

To prove that $$\theta=\frac{\pi}{5}$$ satisfies $$\frac{1}{2}+\cos{(2\theta)}=\cos{\theta}$$:

$$2\sin{\theta}(\frac{1}{2}+\cos{(2\theta)})=2\sin{\theta} \cos{\theta}$$

$$\sin{\theta}+ 2\sin{\theta} \cos{(2\theta)}=2\sin{\theta} \cos{\theta}$$

$$\sin{(2-1)\theta}+2\sin{\theta} \cos{(2\theta)}=\sin{(2\theta)}$$

$$\sin{(2\theta)}\cos{\theta}-\sin{\theta}\cos{(2\theta)}+2\sin{\theta} \cos{(2\theta)}=\sin{(2\theta)}$$

$$\sin{(2\theta)}\cos{\theta}+\sin{\theta}\cos{(2\theta)}=\sin{(2\theta)}$$

$$\sin{(2+1)\theta}=\sin{(2\theta)}$$

$$\sin{(3\theta)}=\sin{(2\theta)}$$

$$\sin{(3\theta)}=\sin{(\pi-2\theta)}$$

$$3\theta=\pi-2\theta$$

$$(3+2)\theta=\pi$$

$$5\theta=\pi$$

$$\theta=\frac{\pi}{5}$$

QED

Note by Noel Lo
3 months, 4 weeks ago

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More generally $$\displaystyle \sum_{k=0}^{n-1} \cos \left( \frac {2k+1}{2n+1}\pi \right) = \frac 12$$ (see proof). This means $$\cos \frac \pi 5 + \cos \frac {3 \pi} 5 = \cos \frac \pi 5 - \cos \frac {2 \pi} 5 = \frac 12$$

- 3 months, 3 weeks ago