Question

The number of all possible values of $\theta ,where0<\theta <\pi ,forwhichthesystemofequations(y+z)cos3\theta =(xyz)sin3\theta xsin3\theta =\frac{2cos3\theta }{y}+\frac{2sin3\theta }{z}and(xyz)sin3\theta =(y+2z)cos3\theta +ysin3\theta haveasolution(x_0,y_0,z_0)with{y}_0{z}_0\ne 0,is.........$

Answer 1

Answer: 3

Given equations can be written as
$xsin3\theta -\frac{cos3\theta }{y}-\frac{cos3\theta }{z}=\mathrm{0...}(i)$
$xsin3\theta -\frac{2cos3\theta }{y}-\frac{2sin3\theta }{z}=\mathrm{0...}(ii)$
$andxsin3\theta -\frac{2}{y}cos3\theta -\frac{1}{z}(cos3\theta +sin3\theta )=\mathrm{0...}(ii)$
Eqs. (ii) and (iii), implies
$2sin3\theta =cos3\theta +sin3\theta \Rightarrow sin3\theta =cos3\theta$
$\therefore tan3\theta =1$
$\Rightarrow 3\theta =\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4}or\theta =\frac{\pi }{12},\frac{5\pi }{12},\frac{9\pi }{12}$