Zadatak 0985

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

Vrednost proizvoda $$\cos \frac{\pi }{7}\cos \frac{4\pi }{7}\cos \frac{5\pi }{7}$$ jednaka je:

a) \(\frac{1}{4}\);          b) \(\frac{\sqrt{2}}{8}\);          c) \(\frac{1}{8}\);          d) \(\frac{\sqrt{2}}{16}\);          e) \(\frac{1}{16}\).

Rešenje:


$$\cos \frac{\pi }{7}\cos \frac{4\pi }{7}\cos \frac{5\pi }{7}=\cos \frac{\pi }{7}\cos \frac{4\pi }{7}\cos (\pi -\frac{2\pi }{7})$$

$$=-\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}=-\frac{2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}}{2\sin \frac{\pi }{7}}$$

$$=-\frac{2\sin \frac{2\pi }{7}\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}}{4\sin \frac{\pi }{7}}=-\frac{2\sin \frac{4\pi }{7}\cos \frac{4\pi }{7}}{8\sin \frac{\pi }{7}}$$

$$=-\frac{\sin \frac{8\pi }{7}}{8\sin \frac{\pi }{7}}=-\frac{(-\sin \frac{\pi }{7})}{8\sin \frac{\pi }{7}}=\frac{1}{8}$$

Tačan odgovor je c).