Question

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

Answer 1

We have,

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

$\tan \theta =\frac{1}{\sqrt{3}}$

$\theta ={30}^0$

Since,

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

$\Rightarrow \frac{3{\cos }^2{30}^0+2\cos {30}^0}{3\cos {30}^0+2}$

$\Rightarrow \frac{3{\left(\frac{\sqrt{3}}{2}\right)}^2+2\left(\frac{\sqrt{3}}{2}\right)}{3\left(\frac{\sqrt{3}}{2}\right)+2}$

$\Rightarrow \frac{\frac{9}{4}+\sqrt{3}}{\frac{3\sqrt{3}+4}{2}}$

$\Rightarrow \frac{\frac{4\sqrt{3}+9}{4}}{\frac{3\sqrt{3}+4}{2}}$

$\Rightarrow \frac{4\sqrt{3}+9}{2(3\sqrt{3}+4)}$

Hence, this is the answer.