The value of theta satisfying the given equat | vedclass

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Question

(A) Given equation: $\cos 	heta + \sqrt{3} \sin 	heta = 2$. Divide the entire equation by $2$: $\frac{1}{2} \cos 	heta + \frac{\sqrt{3}}{2} \sin

Vedclass · Accepted Answer

$\frac{\pi}{3}$

Vedclass · Answer

(A) Given equation: $\cos 	heta + \sqrt{3} \sin 	heta = 2$. Divide the entire equation by $2$: $\frac{1}{2} \cos 	heta + \frac{\sqrt{3}}{2} \sin 	heta = 1$. Using the identity $\sin(A + B) = \sin A \cos B + \cos A \sin B$,we can rewrite this as: $\sin 	heta \cos \frac{\pi}{3} + \cos 	heta \sin \frac{\pi}{3} = 1$. This simplifies to: $\sin(	heta + \frac{\pi}{3}) = 1$. Since $\sin(\frac{\pi}{2}) = 1$,we have: $	heta + \frac{\pi}{3} = \frac{\pi}{2}$. $	heta = \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6}$. Wait,checking the options: If $	heta = \frac{\pi}{3}$,then $\cos(\frac{\pi}{3}) + \sqrt{3} \sin(\frac{\pi}{3}) = \frac{1}{2} + \sqrt{3}(\frac{\sqrt{3}}{2}) = \frac{1}{2} + \frac{3}{2} = 2$. Thus,$	heta = \frac{\pi}{3}$ is the correct solution.

The value of $\theta$ satisfying the given equation $\cos \theta + \sqrt{3} \sin \theta = 2$ is:

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