Question

If $2\sin \theta +\tan \theta =0$, then the general values of $\theta$ are

• A. $2n\pi \pm \frac{\pi }{3}$
• B. $n\pi ,2n\pi \pm \frac{2\pi }{3}$
• C. $n\pi ,2n\pi \pm \frac{\pi }{3}$
• D. $n\pi ,n\pi +\frac{2\pi }{3}$

Answer 1

The correct option is B

$n\pi ,2n\pi \pm \frac{2\pi }{3}$

Explanation for the correct option:

Find the value of $\theta$:

Given,

$2\sin \theta +\tan \theta =0$

$\Rightarrow$ $2\sin \theta +\frac{\sin \theta }{\cos \theta }=0$

$\Rightarrow$$2\sin \theta \cos \theta +\sin \theta =0$

$\Rightarrow$ $\sin \theta (2\cos \theta +1)=0$

$\Rightarrow$$\sin \theta =0,2\cos \theta +1=0$

$\Rightarrow$$\sin \theta =0,\cos \theta =-1/2$

$\Rightarrow$$\theta =n\pi ,\cos \theta =\cos \left(\frac{2\mathrm{\pi }}{3}\right)$

$\therefore \mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathit{\pi }\mathbf{,}\mathbf{}\mathbf{2}\mathit{n}\mathit{\pi }\mathbf{}\mathbf{\pm }\mathbf{}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}$

Hence, Option ‘B’ is Correct.