Question

If $2\sin \alpha +3\cos \alpha =2$, then $3\sin \alpha -2\cos \alpha =$______?

• A. $\pm 3$
• B. $\pm 1$
• C. $0$
• D. $\pm 2$

Answer 1

The correct option is A $\pm 3$

Let $3\sin \alpha -2\cos \alpha =t$ ...(1)

Given : $2\sin \alpha +3\cos \alpha =2$ ...(2)

$⟹{(3\sin \alpha -2\cos \alpha )}^2+{(2\sin \alpha +3\cos \alpha )}^2=t^2+2^2$

$⟹9{\sin }^2\alpha -12\sin \alpha \cos \alpha +4{\cos }^2\alpha +4{\sin }^2\alpha +12\sin \alpha \cos \alpha +9{\cos }^2\alpha =t^2+4$

$⟹13({\sin }^2\alpha +{\cos }^2\alpha )=t^2+4$

$⟹$ $t^2=9$

$⟹$ $t=\pm 3$