The number of solutions of cos 2 theta = sin | vedclass

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Question

(A) Given equation: $\cos 2 	heta = \sin 	heta$ Using the identity $\cos 2 	heta = 1 - 2 \sin^2 	heta$,we get: $1 - 2 \sin^2 	heta = \sin 	heta$

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(A) Given equation: $\cos 2 	heta = \sin 	heta$ Using the identity $\cos 2 	heta = 1 - 2 \sin^2 	heta$,we get: $1 - 2 \sin^2 	heta = \sin 	heta$ $2 \sin^2 	heta + \sin 	heta - 1 = 0$ Factoring the quadratic equation: $(2 \sin 	heta - 1)(\sin 	heta + 1) = 0$ This gives $\sin 	heta = \frac{1}{2}$ or $\sin 	heta = -1$. For $\sin 	heta = \frac{1}{2}$ in $(0, 2 \pi)$,$	heta = \frac{\pi}{6}$ and $	heta = \frac{5 \pi}{6}$. For $\sin 	heta = -1$ in $(0, 2 \pi)$,$	heta = \frac{3 \pi}{2}$. Thus,the solutions are $\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$. The total number of solutions is $3$.

The number of solutions of $\cos 2 \theta = \sin \theta$ in the interval $(0, 2 \pi)$ is:

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