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  4. If tan θ+ tan 2 θ+√3 tan θ tan 2 θ=√3, then the general values of θ are

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Q. If $\tan \theta+\tan 2 \theta+\sqrt{3} \tan \theta \tan 2 \theta=\sqrt{3}$, then the general values of $\theta$ are

AP EAMCETAP EAMCET 2019

Solution:

It is given that
$\tan \,\theta+\tan \,2 \, \theta+\sqrt{3} \,\tan \,\theta \,\tan \,2 \,\theta=\sqrt{3}$
$\Rightarrow \frac{\tan \,\theta+\tan 2 \,\theta}{1-\tan \,\theta \,\tan \,2 \,\theta}=\sqrt{3}=\tan \frac{\pi}{3}$
$\Rightarrow \tan (3 \,\theta)=\tan \frac{\pi}{3}$
$ \Rightarrow 3 \, \theta=n \pi+\frac{\pi}{3}, n \in Z$
$\Rightarrow \theta=(3 n+1) \frac{\pi}{9}, n \in Z$