If sin 2theta = cos 3theta and theta is an ac | vedclass

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(A) Given $\sin 2	heta = \cos 3	heta$. We can write this as $\sin 2	heta = \sin(\frac{\pi}{2} - 3	heta)$. General solution for $\sin x = \sin y$ i

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Given $\sin 2	heta = \cos 3	heta$. We can write this as $\sin 2	heta = \sin(\frac{\pi}{2} - 3	heta)$. General solution for $\sin x = \sin y$ is $x = n\pi + (-1)^n y$. Case $1$: $2	heta = 2n\pi + (\frac{\pi}{2} - 3	heta)$ $\Rightarrow 5	heta = 2n\pi + \frac{\pi}{2}$ $\Rightarrow 	heta = \frac{2n\pi}{5} + \frac{\pi}{10}$. For $n=0$,$	heta = \frac{\pi}{10} = 18^\circ$. Case $2$: $2	heta = (2n+1)\pi - (\frac{\pi}{2} - 3	heta)$ $\Rightarrow 2	heta = 2n\pi + \pi - \frac{\pi}{2} + 3	heta$ $\Rightarrow 	heta = -2n\pi - \frac{\pi}{2}$. Since $	heta$ is an acute angle,we take $	heta = 18^\circ$. Then $\sin 18^\circ = \frac{\sqrt{5} - 1}{4}$.

If $\sin 2\theta = \cos 3\theta$ and $\theta$ is an acute angle,then $\sin \theta$ is equal to

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