Question

Demonstrați identitatea de ma jos:
cos((2*pi)/5)+cos((4*pi)/5)= -1/2
(Aveti poza mai jos)
Multumesc!!!

Answer 1

$E = \cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5} \Big|\cdot \sin\dfrac{\pi}{5}\\ \\ E\cdot\sin \dfrac{\pi}{5} = \sin \dfrac{\pi}{5} \cos\dfrac{2\pi}{5}+ \sin \dfrac{\pi}{5}\cos\dfrac{4\pi}{5} \\ \\ E\cdot \sin \dfrac{\pi}{5} = \dfrac{\sin\Big(-\dfrac{\pi}{5}\Big)+\sin \dfrac{3\pi}{5}}{2} +\dfrac{\sin\Big(-\dfrac{3\pi}{5}\Big)+\sin\pi }{2} \\ \\ 2E\cdot \sin \dfrac{\pi}{5} = -\sin \dfrac{\pi}{5}+\sin \dfrac{3\pi}{5}-\sin \dfrac{3\pi}{5}+0 \\\\ 2E\cdot \sin \dfrac{\pi}{5} = - \sin \dfrac{\pi}{5} \\ \\ 2E = -1$

$\Rightarrow E = -\dfrac{1}{2}$