Question

If $3\sin \theta +5\cos \theta =5$, show that
$5\sin \theta -3\cos \theta =\pm 3$.

Answer 1

Squaring $3\sin \theta +5\cos \theta =5$, we get
$9{\sin }^2\theta +25{\cos }^2\theta +30\sin \theta \cos \theta =25$
or $9(1-{\cos }^2\theta )+25(1-{\sin }^2\theta )+30\sin \theta \cos \theta =25$
or $9=9{\cos }^2\theta +25{\sin }^2\theta -30\sin \theta \cos \theta$
$={(5\sin \theta -3\cos \theta )}^2$
$\therefore 5\sin \theta -3\cos \theta =\pm 3$.