Question : If $3\sin \theta +5\cos \theta =5$, then $5\sin \theta -3\cos \theta$ is equal to:
Option 1: $\pm 3$
Option 2: $\pm 5$
Option 3: $1$
Option 4: $\pm 2$
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Correct Answer: $\pm 3$
Solution :
Given,
$3\sin\theta + 5\cos\theta =5$
squaring both sides, we get,
$⇒9\sin^2\theta + 25\cos^2\theta+30\sin\theta \cos\theta =25$
$⇒ 30\sin\theta \cos\theta =25- 9\sin^2\theta - 25\cos^2\theta$--------------------------(1)
To find $5\sin\theta-3\cos\theta$,
$(5\sin\theta-3\cos\theta)^2 = 25\sin^2\theta+9\cos^2\theta-30\sin\theta \cos\theta$
Using equation(1), we get,
$(5\sin\theta-3\cos\theta)^2 = 25\sin^2\theta+9\cos^2\theta-(25-9\sin^2\theta-25\cos^2\theta)$
$⇒ (5\sin\theta-3\cos\theta)^2 = 25\sin^2\theta+9\cos^2\theta-25+9\sin^2\theta+25\cos^2\theta$
$⇒ (5\sin\theta-3\cos\theta)^2 = 34\sin^2\theta+34\cos^2\theta-25$
$⇒ (5\sin\theta-3\cos\theta)^2 = 34(\sin^2\theta+\cos^2\theta)-25$
$⇒ (5\sin\theta-3\cos\theta)^2 = 34(1)-25$
$⇒ (5\sin\theta-3\cos\theta)^2 = 9$
$\therefore(5\sin\theta-3\cos\theta) = \pm 3$
Hence, the correct answer is $\pm3$.
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