Trigonometrical Ratios and Identities 1 Question 29
29. The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations
$$ \begin{aligned} (y+z) \cos 3 \theta & =(x y z) \sin 3 \theta \\ x \sin 3 \theta & =\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z} \end{aligned} $$
and $\quad(x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta$ have a solution $\left(x _0, y _0, z _0\right)$ with $y _0 z _0 \neq 0$, is ……
(2010)
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Solution:
- Given equations can be written as
$$ \begin{aligned} x \sin 3 \theta-\frac{\cos 3 \theta}{y}-\frac{\cos 3 \theta}{z} & =0 \\ x \sin 3 \theta-\frac{2 \cos 3 \theta}{y}-\frac{2 \sin 3 \theta}{z} & =0 \end{aligned} $$
and $x \sin 3 \theta-\frac{2}{y} \cos 3 \theta-\frac{1}{z}(\cos 3 \theta+\sin 3 \theta)=0$
Eqs. (ii) and (iii), implies
$$ \begin{array}{rlrl} & & 2 \sin 3 \theta & =\cos 3 \theta+\sin 3 \theta \\ & \therefore & \sin 3 \theta & =\cos 3 \theta \\ \Rightarrow & \tan 3 \theta & =1 \\ & & 3 \theta & =\frac{\pi}{4}, \frac{5 \pi}{4}, \frac{9 \pi}{4} \\ \text { or } & & \theta & =\frac{\pi}{12}, \frac{5 \pi}{12}, \frac{9 \pi}{12} \end{array} $$
