IMO 1963 #5

Database Soal Olimpiade Matematika dari Seluruh Dunia

5. Buktikan bahwa $\cos\frac{\pi}7-\cos\frac{2\pi}7+\cos\frac{3\pi}7=\frac12$ .

Solusi:

Ruas kiri sama dengan $S=\cos\frac{\pi}7+\cos\frac{3\pi}7+\cos\frac{5\pi}7$ . Maka

$S\sin\frac{\pi}7=\frac{\sin(2\pi/7)}2+\frac{\sin(4\pi/7)-\sin(2\pi/7)}2+\frac{\sin(6\pi/7)-\sin(4\pi/7)}2=\frac{\sin(6\pi/7)}2=\frac{\sin(\pi/7)}2.$

Jadi $S=\frac12$ .

Written by olimpiadematematika

12 April 2009 pada 15:18

Olimpiade Matematika