The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations…
The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations
$\begin{array}{l}
(y+z) \cos 3 \theta=(x y z) \sin 3 \theta \\
x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z}
\end{array}$
and $(x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$
have a solution $\left(x_0, y_0, z_0\right)$ with $\quad y_0 z_0 \neq 0$, is
Solution
Given equations can be written as
$
\begin{gathered}
x \sin 3 \theta-\frac{\cos 3 \theta}{y}-\frac{\cos 3 \theta}{z}=0 \\
x \sin 3 \theta-\frac{2 \cos 3 \theta}{y}-\frac{2 \sin 3 \theta}{z}=0 \\
\text { and } x \sin 3 \theta-\frac{2}{y} \cos 3 \theta \\
-\frac{1}{z}(\cos 3 \theta+\sin 3 \theta)=0
\end{gathered}
$
Eqs. (ii) and (iii), implies
$
\begin{aligned}
& 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} \text { or } \theta=\frac{\pi}{12}, \frac{5 \pi}{12}, \frac{9 \pi}{12} .
\end{aligned}
$