Question

If $3\sin \theta +5\cos \theta =5$, then $5\sin \theta -3\cos \theta$ is equal to

• A. $3$
• B. $8$
• C. $13$
• D. none of these

Answer 1

Answer: (A)

$3$

Given $3sin\theta +5cos\theta =5$
Squaring both sides, we get
$9{\sin }^2\theta +25{\cos }^2\theta +30\sin \theta \cos \theta =25$
$\Rightarrow 9-9{\cos }^2\theta +25-25{\sin }^2\theta +30\sin \theta \cos \theta =25$
$\Rightarrow 9{\cos }^2\theta +25{\sin }^2\theta -30\sin \theta \cos \theta =9$
$\Rightarrow {(5\sin \theta -3\cos \theta )}^2=9$
$\Rightarrow 5\sin \theta -3\cos \theta =\pm 3$