Question

The number of distinct solutions of $\sin 5\theta \,\cos\, 3\theta= \sin \,9\theta \,\cos 7\theta\, in \left[0,\frac{\pi}{2}\right] $ is

• A. 4
• B. 5
• C. 8
• D. 9

Answer 1

Answer: (D)

$\sin 5\theta\, \cos 3\theta = \sin 9\theta \cos 7\theta$
$\Rightarrow \:\: 2 \sin 5\theta \cos 3\theta = 2 \sin 9\theta \cos 7\theta$
$\Rightarrow \:\: \sin 8\theta + \sin 2\theta = \sin 16\theta + \sin 2\theta$
$\Rightarrow \:\: \sin 16\theta = \sin 8\theta \Rightarrow 16\theta = n\pi+(-1)^n 8\theta$
$\Rightarrow $ 16$\theta = 2m\pi+8\theta$
[If $n$ is even and $n = 2m$]
$\Rightarrow \:\: 8\theta =2m\pi\, \Rightarrow \theta=\frac{m\pi}{4}$
and $16\theta = (2m+1)\pi - 8\theta$
[If $n$ is odd and $n = 2m + 1$]
$\Rightarrow $ $24\theta = (2m+1)\pi \, \Rightarrow \, \theta=\frac{2m+1}{24}\pi$
Thus, $\theta=\frac{m\pi}{4},\frac{2m+1}{24}\pi$ where $m\, \in\,Z$
$= 0 ,\frac{\pi}{4},\frac{\pi}{2},\frac{\pi}{24},\frac{7\pi}{24},\frac{3\pi}{24},\frac{11\pi}{24}$
$\therefore $ total no. of solutions = 9