Question

If $\sqrt{3}\sin \theta =\cos \theta$, find the value of $\frac{3{\cos }^2\theta +2\cos \theta }{3\cos \theta +2}$

Answer 1

Given ,$\sqrt{3}\sin \theta =\cos \theta$

$\Rightarrow \frac{\sin \theta }{\cos \theta }=\frac{1}{\sqrt{3}}$

$\text{or},\tan \theta =\frac{1}{\sqrt{3}}$

$\Rightarrow \tan \theta =\tan {30}^{\circ }$ $[\because \tan {30}^{\circ }=\frac{1}{\sqrt{3}}]$

$\Rightarrow \theta ={30}^{\circ }$

On simplifying the given equation, we get

$\frac{3{\cos }^2\theta +2\cos \theta }{3\cos \theta +2}$

$=\frac{\cos \theta (3\cos \theta +2)}{(3\cos \theta +2)}$ [Taking $\cos \theta$ common in the numerator]

$=\cos \theta$

Substituting $\theta ={30}^{\circ }$, we get

$=\cos {30}^{\circ }$

$=\frac{\sqrt{3}}{2}$

$\therefore \frac{3{\cos }^2\theta +2\cos \theta }{3\cos \theta +2}=\frac{\sqrt{3}}{2}$