Question

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

• A. $-9$
• B. $9$
• C. $3$
• D. $-3$

Answer 1

Answer: (C), (D)

$-3$

Given that $3\sin \theta +5\cos \theta =5,$
And let $5\sin \theta -3\cos \theta =x$
Squaring and adding both the equations, we get,
$9{\sin }^2\theta +25{\cos }^2\theta +30\sin \theta \cos \theta$
$+25{\sin }^2\theta +9{\cos }^2\theta -30\sin \theta \cos \theta =25+x^2$
Since ${\sin }^2\theta +{\cos }^2\theta =1$ ,we get
$\Rightarrow 9+25=25+x^2$
$\Rightarrow x^2=9\Rightarrow x=-3,3$