Question

If $\sin 3\theta =\cos 2\theta$ then $\theta$ is equal to

• A. $(4n+1)\frac{\pi }{2},(4n+1)\frac{\pi }{10};$ when n is even integer only
• B. $(4n+1)\frac{\pi }{2},(4n+1)\frac{\pi }{10};$ when n is an any integer
• C. $(4n+1)\frac{\pi }{2},(4n+1)\frac{\pi }{10};$ when n is an odd integer only
• D. none of these

Answer 1

Answer: (B)

$(4n+1)\frac{\pi }{2},(4n+1)\frac{\pi }{10};$ when n is an any integer

Given, $\sin 3\theta =\cos 2\theta$

$\Rightarrow \cos (\frac{\pi }{2}-3\theta )=\cos 2\theta$

General solution is,
$\frac{\pi }{2}-3\theta =2n\pi \pm 2\theta$

Taking (+) sign$\frac{\pi }{2}-3\theta =2n\pi +2\theta$
$5\theta =(-4n+1)\frac{\pi }{2}$

$\Rightarrow \theta =(4m+1)\frac{\pi }{10}$ where $m=-n$
Taking (-) sign
$\frac{\pi }{2}-3\theta =2n\pi -2\theta$

$\Rightarrow \theta =(-4n+1)\frac{\pi }{2}$

$\Rightarrow \theta =(4m+1)\frac{\pi }{2}$ where $m=-n$