question_answer

If \[\sqrt{3}\text{ }sin\text{ }\theta =cos\text{ }\theta \], find the value of \[\frac{3\,{{\cos }^{2}}\theta +2\,\cos \theta }{3\,\cos \theta +2}\].

Answer:

\[\sqrt{3}\text{ }sin\text{ }\theta =cos\text{ }\theta \]             [Given]

\[\Rightarrow \frac{\sin \theta }{\cos \theta }=\frac{1}{\sqrt{3}}\] or \[\tan \,\theta =\frac{1}{\sqrt{3}}\]

\[\Rightarrow \tan \theta =\tan 30{}^\circ \Rightarrow \theta =30{}^\circ \]

Now,

\[\frac{3\,{{\cos }^{2}}\theta +2\cos \theta }{3\cos \theta +2}=\frac{\cos \theta (3cos\theta +2)}{(3cos\theta +2)}\]

\[=\cos \theta \]

Put                   \[\theta =30{}^\circ \]

\[\Rightarrow \cos 30{}^\circ =\frac{\sqrt{3}}{2}\]