2nd VMO 1963 Problem 3 Solve the equation sin 3 x cos 3x + cos 3 x sin 3x = 3/8.
7½ o + k90 o or 37½ o + k90 o Solution We have sin 3x = 3 sin x - 4 sin 3 x, cos 3x = 4 cos 3 x - 3 cos x. So we need 4 sin 3 x cos 3 x - 3 sin 3 x cos x + 3 sin x cos 3 x - 4 sin 3 x cos 3 x = 3/8 or 8 sin x cos x(cos 2 x - sin 2 x) = 1, or 4 sin 2x cos 2x = 1 or sin 4x = 1/2. Hence 4x = 30 o + k360 o or 150 o + k360 o . So x = 7½ o + k90 o or 37½ o + k90 o . Thanks to Suat Namli 2nd VMO 1963 © John Scholes jscholes@kalva.demon.co.uk 7 March 2004 Last corrected 7 Mar 04