Question

lf $0<\theta <\frac{\pi }{2}$ , $\sin 2\theta =\cos 3\theta$, then $\sin \theta$ is equal to

• A. $\frac{\sqrt{5}-1}{4}$
• B. $\frac{\sqrt{5}+1}{4}$
• C. $0$
• D. $\frac{\sqrt{10-2\sqrt{5}}}{4}$

Answer 1

Answer: (A)

$\frac{\sqrt{5}-1}{4}$

Given:

$\sin 2\theta =\cos 3\theta$
$\Rightarrow 2\sin \theta \cos \theta =4{\cos }^3\theta -3\cos \theta$
Cancelling $\cos \theta$ we get,
$2\sin \theta =4{\cos }^2\theta -3$
$2\sin \theta =1-4{\sin }^2\theta$
So $\sin \theta =\frac{\sqrt{5}-1}{4}$