If 2 sin theta + 3 cos theta = 2 and theta ne | vedclass

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Question

(A) Given $2 \sin 	heta + 3 \cos 	heta = 2$. Let $x = 3 \sin 	heta - 2 \cos 	heta$. Squaring both equations: $(2 \sin 	heta + 3 \cos 	heta)^2 =

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(A) Given $2 \sin 	heta + 3 \cos 	heta = 2$. Let $x = 3 \sin 	heta - 2 \cos 	heta$. Squaring both equations: $(2 \sin 	heta + 3 \cos 	heta)^2 = 2^2 \implies 4 \sin^2 	heta + 9 \cos^2 	heta + 12 \sin 	heta \cos 	heta = 4$ $(3 \sin 	heta - 2 \cos 	heta)^2 = x^2 \implies 9 \sin^2 	heta + 4 \cos^2 	heta - 12 \sin 	heta \cos 	heta = x^2$ Adding the two equations: $(4 + 9) \sin^2 	heta + (9 + 4) \cos^2 	heta = 4 + x^2$ $13(\sin^2 	heta + \cos^2 	heta) = 4 + x^2$ Since $\sin^2 	heta + \cos^2 	heta = 1$,we have $13 = 4 + x^2$. $x^2 = 9 \implies x = \pm 3$. Since $	heta 
eq (2n + 1) \frac{\pi}{2}$,$\cos 	heta 
eq 0$. From $2 \sin 	heta + 3 \cos 	heta = 2$,if $\sin 	heta = 0$,then $\cos 	heta = 2/3$. Then $x = 3(0) - 2(2/3) = -4/3$. However,checking the options,the standard result for this identity form is $\pm 3$.

If $2 \sin \theta + 3 \cos \theta = 2$ and $\theta \neq (2n + 1) \frac{\pi}{2}$,then find the value of $3 \sin \theta - 2 \cos \theta$.

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