Iloczyn sinusów

dwukwiat15 · Post autor: **dwukwiat15** » 28 gru 2006, o 01:23

Udowodnij że:

cos(pi/5) * cos(2pi/5) = 1/4

wb · Post autor: wb » 28 gru 2006, o 09:03

\(\displaystyle{ cos\frac{\pi}{5}\cdot cos\frac{2\pi}{5}=\frac{2sin\frac{\pi}{5}\cdot cos\frac{\pi}{5}\cdot \frac{2\pi}{5}}{2sin\frac{\pi}{5}}=\frac{sin\frac{2\pi}{5}\cdot cos\frac{2\pi}{5}}{2sin\frac{\pi}{5}}= \\ =\frac{2sin\frac{2\pi}{5}cos\frac{2\pi}{5}}{4sin\frac{\pi}{5}}=\frac{sin\frac{4\pi}{5}}{4sin\frac{\pi}{5}}=\frac{sin(\pi - \frac{\pi}{5})}{4sin\frac{\pi}{5}}=\frac{sin\frac{\pi}{5}}{4sin\frac{\pi}{5}}=\frac{1}{4}}\)