Zadatak 0960

Priprema za prijemni za "tehničke" fakultete (ETF, MatF, MašF, Fon, SF, TMF,...)

Vrednost izraza $$\cos \frac{\pi }{5}\cdot \cos \frac{2\pi }{5}$$ je:

a) \(\frac{1}{16}\);          b) \(\frac{1}{2}\);          c) \(\frac{1}{3}\);          d) \(\frac{\sqrt{3}}{16}\);          e) \(\frac{1}{4}\).

Rešenje:


$$\cos \frac{\pi }{5}\cdot \cos \frac{2\pi }{5}=\frac{2\sin \frac{\pi }{5}\cos \frac{\pi }{5}\cos \frac{2\pi }{5}}{2\sin \frac{\pi }{5}}$$

$$=\frac{2\sin \frac{2\pi }{5}\cos \frac{2\pi }{5}}{4\sin \frac{\pi }{5}}$$

$$=\frac{\sin \frac{4\pi }{5}}{4\sin \frac{\pi }{5}}$$

$$=\frac{\sin \left ( \pi -\frac{\pi }{5} \right )}{4\sin \frac{\pi }{5}}$$

$$=\frac{\sin \frac{\pi }{5}}{4\sin \frac{\pi }{5}}=\frac{1}{4}$$

Tačan odgovor je e).