Zadatak 0995

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

Zbir $$\tan 9^{\circ}+\tan 81^{\circ}+\tan 117^{\circ}+\tan 153^{\circ}$$ je jednak:

a) \(-\frac{13\sqrt{3}}{5}\);          b) \(-3\);         c) \(1\);         d) \(4\);         e) \(3\sqrt{3}\).

Rešenje:


$$\tan 9^{\circ}+\tan 81^{\circ}+\tan 117^{\circ}+\tan 153^{\circ}=(\tan 9^{\circ}+\tan 81^{\circ})-(\tan 63^{\circ}+\tan 27^{\circ})$$

$$=\frac{\sin 9^{\circ}}{\cos 9^{\circ}}+\frac{\sin 81^{\circ}}{\cos 81^{\circ}}-\left ( \frac{\sin 63^{\circ}}{\cos 63^{\circ}} +\frac{\sin 27^{\circ}}{\cos 27^{\circ}}\right )$$

$$=\frac{\sin (9^{\circ}+81^{\circ})}{\cos 9^{\circ}\cos 81^{\circ}}-\frac{\sin (27^{\circ}+63^{\circ})}{\cos 27^{\circ}\cos 63^{\circ}}$$

$$=\frac{2}{\sin 18^{\circ}}-\frac{2}{\sin 54^{\circ}}=\frac{2(\sin 54^{\circ}-\sin 18^{\circ})}{\sin 18^{\circ}\sin 54^{\circ}}$$

$$=\frac{4\cos 36^{\circ}\sin 18^{\circ}}{\sin 18^{\circ}\sin 54^{\circ}}=4$$

Tačan odgovor je d).