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} $
$\begin{bmatrix}\because \sin\theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\\ \text{and} 1-\cos\theta = 2\sin^{2} \frac{\theta}{2}\end{bmatrix} $
$\Rightarrow \tan \frac{\theta}{2} = \frac{3}{5}$
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. $