Question

If $3\cos \theta +4\sin \theta =A\sin (\theta +\alpha ),$ then values of $A$ and $\alpha$ are

• A. $5,{\cos }^{-1}\left(\frac{3}{5}\right)$
• B. $5,{\sin }^{-1}\left(\frac{3}{5}\right)$
• C. $7,{\sin }^{-1}\left(\frac{4}{5}\right)$
• D. $Noneofthese$

Answer 1

The correct option is B $5,{\sin }^{-1}\left(\frac{3}{5}\right)$

$3\cos \theta +4\sin \theta =A\sin (\theta +\alpha )$

LHS:

$=3\cos \theta +4\sin \theta$

$=\sqrt{3^2+4^2}[\frac{3}{\sqrt{3^2+4^2}}\cos \theta +\frac{4}{\sqrt{3^2+4^2}}\sin \theta ]$

$=5[\frac{3}{5}\cos \theta +\frac{4}{5}\sin \theta ]$

Let, $\sin \alpha =\frac{3}{5},\cos \alpha =\frac{4}{5}$

$=5[\sin \alpha \cos \theta +\cos \alpha \sin \theta ]$

$=5\sin (\theta +\alpha )$

comparing the above equation with RHS.

$A=5$

$\alpha ={\sin }^{-1}\frac{3}{5}$