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)

Given, $3\sin \theta +5\cos \theta =5$
$\Rightarrow 3\sin \theta =5\left(1-\cos \theta \right)$
$\Rightarrow 3.2\sin \frac{\theta }{2}\cos \frac{\theta }{2}=5.2{\sin }^2\frac{\theta }{2}$
$\left[\begin{array}{c}\because \sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\\ \text{and}1-\cos \theta =2{\sin }^2\frac{\theta }{2}\end{array}\right]$
$\Rightarrow \tan \frac{\theta }{2}=\frac{3}{5}$
Now , $5\sin \theta -3\cos \theta$
$=5.\frac{2\tan \frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}-3.\frac{1-{\tan }^2\frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}$
$=5.\frac{2.\frac{3}{5}}{\left(1+\frac{9}{25}\right)}-3.\frac{\left(1-\frac{9}{25}\right)}{\left(1+\frac{9}{25}\right)}$
$=\frac{6-3.\frac{16}{25}}{1+\frac{9}{25}}$
$=\frac{150-48}{34}=\frac{102}{34}=3.$