The number of all possible values of `theta,` where, for which the system of equations `(y+z)cos3theta=(x y z)sin3theta` `xsin3theta=(2cos3theta)/y+(2sin3theta)/z` `(x y z)sin3theta=(y+2z)cos3theta+ysin3theta` has a solution `(x_0,y_0,z_0)` with `y_0z_0!=0` is

Answer

3

Explanation

Given equations can be written as
`xsin3 theta-(cos 3 theta)/(y)-(cos3theta)/(z) = 0 " ...(i)"`
`xsin 3 theta - (2 cos 3 theta)/(y) - (2sin 3theta)/(z) = 0 " ...(ii)"`
and `x sin 3 theta - (2)/(y) cos 3 theta -(1)/(z) (cos 3 theta + sin 3 theta)=0 " ...(iii)"`
Eqs. (ii) and (iii), implies
`2 sin 3 theta= cos 3 theta + sin 3 theta`
`implies sin 3 theta = cos 3 theta`
`therefore tan 3 theta = 1`
`implies 3 theta = (pi)/(4),(5pi)/(4),(9pi)/(4)`
`"or " theta=(pi)/(12),(5pi)/(12),(9pi)/(12)`