Question

If $4\sin \theta +3\cos \theta =5$ then $\tan \theta =$

• A. $\frac{3}{4}$
• B. $\frac{3}{5}$
• C. $\frac{4}{3}$
• D. $\frac{4}{5}$

Answer 1

Answer: (C)

$\frac{4}{3}$

Given $4\sin \theta +3\cos \theta =5$
Dividing the entire expression by $5$,
$\frac{4}{5}\sin \theta +\frac{3}{5}\cos \theta =1$
Let, $\frac{4}{5}=\cos A$, so $\frac{3}{5}=\sin A$
$\cos A\sin \theta +\sin A\cos \theta =\sin \left(\frac{\pi }{2}\right)$
$\sin (A+\theta )=\sin \left(\frac{\pi }{2}\right)$
$\theta =\frac{\pi }{2}-A$
So $\tan \theta =\cot A$
$\cot A=\frac{\cos A}{\sin A}=\frac{4}{3}=\tan \theta$