Question

If $3\sin \theta +5\cos \theta =5$, then the value(s) of $5\sin \theta -3\cos \theta$ is/are:

• A.
  2
• B.
  3
• C.
  -2
• D.
  -3

Answer 1

Answer: (B), (D)

-3
$\mathrm{Given}3\sin \theta +5\cos \theta =5$
Now, if we assume
$5\sin \theta -3\cos \theta =x$
Squaring and adding both equations we get:
$\left(9{\sin }^2\theta +{\mathrm{25cos}}^2\theta +30\sin \theta \cos \theta \right)+(25{\sin }^2\theta +9{\cos }^2\theta -30\sin \theta \cos \theta )=25+x^2$
$\Rightarrow 9({\sin }^2\theta +{\cos }^2\theta )+25({\sin }^2\theta +{\cos }^2\theta )=25+x^2$
If we use the trigonometric identity: $({\cos }^2\theta +{\sin }^2\theta )=1$ in the equation, we get our equation as:
$9+25=25+x^2\Rightarrow 9=x^2\Rightarrow x=\pm 3$
$\Rightarrow 5\sin \theta -3\cos \theta =\pm 3$
Thus, the options b and d are correct.