Question

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

• A. $5$
• B. $3$
• C. $4$
• D. None of these

Answer 1

Answer: (B)

$3$

Let $5\sin \theta -3\cos \theta =k\cdots \left(1\right)$

Given $3\sin \theta +5\cos \theta =5\cdots \left(2\right)$

$(1{)}^2+(2{)}^2⟹(5\sin \theta -3\cos \theta {)}^2+(5\cos \theta +3\sin \theta {)}^2=k^2+5^2$

$⟹25{\sin }^2\theta +9{\cos }^2\theta -30\sin \theta \cos \theta +25{\cos }^2\theta +9{\sin }^2\theta +30\sin \theta \cos \theta =k^2+25$

$⟹25({\sin }^2\theta +{\cos }^2\theta )+9({\sin }^2\theta +{\cos }^2\theta )=k^2+25$

$⟹k^2=25+9-25=9⟹k=\pm 3$

So $5\sin \theta -3\cos \theta =3$