Question

The number of all possible values of $\theta, where 0 < \theta < \pi,for which the system of equations ( y + z) cos 3\theta= (xyz) sin 3 \theta xsin 3\theta=\frac{2 cos 3\theta}{y}+\frac{2 sin 3 \theta}{z} and (xyz) sin 3\theta=(y+2z)cos3\theta +ysin 3\theta have a solution (x_0, y_0, z_0) with y_0 z_0 \ne 0, is .........$

Answer 1

Given equations can be written as
$x sin 3\theta -\frac{cos 3\theta}{y}-\frac{cos 3\theta}{z}=0 ...(i)$
$x sin 3\theta -\frac{2cos 3\theta}{y}-\frac{2sin 3\theta}{z}=0 ...(ii)$
$and xsin 3\theta -\frac{2}{y} cos 3 \theta -\frac{1}{z} (cos 3\theta+sin 3\theta)=0 ...(ii)$
Eqs. (ii) and (iii), implies
$2 sin 3 \theta = cos 3\theta+ sin 3 \theta \Rightarrow sin 3\theta = cos 3 \theta $
$\therefore tan 3\theta = 1$
$\Rightarrow 3\theta=\frac{\pi}{4},\frac{5\pi}{4},\frac{9\pi}{4} or \theta=\frac{\pi}{12},\frac{5\pi}{12},\frac{9\pi}{12}$