Question

If $\frac{\tan 3\theta -1}{\tan 3\theta +1}=\sqrt{3}$, then the general value of $\theta$ is:

• A. $\frac{n\pi }{3}-\frac{\pi }{12},n\in Z$
• B. $n\pi +\frac{7\pi }{12},n\in Z$
• C. $\frac{n\pi }{3}+\frac{7\pi }{36},n\in Z$
• D. $n\pi +\frac{\pi }{12},n\in Z$

Answer 1

The Correct Option is C

Explanation:
Given: $\frac{\tan 3\theta -1}{\tan 3\theta +1}=\sqrt{3}$
We have to find the general value of $\theta$.
Consider, $\frac{\tan 3\theta -1}{\tan 3\theta +1}=\sqrt{3}$$\Rightarrow \frac{\tan 3\theta -\tan \frac{\pi }{4}}{1+\tan 3\theta \cdot \tan \frac{\pi }{4}}=\sqrt{3}$[Using trigonometric ratios ]$$
⇒tan⁡(3θ−π4)=tan⁡π3[Using trigonometric formula of tangent function ]$\Rightarrow 3\theta -\frac{\pi }{4}=n\pi +\frac{\pi }{3},n\in Z$[Using general solution of tangent function]$\Rightarrow 3\theta =n\pi +\frac{\pi }{3}+\frac{\pi }{4},\mathrm{n}\in \mathrm{Z}$$\Rightarrow 3\theta =n\pi +\frac{4\pi +3\pi }{12},\mathrm{n}\in \mathrm{Z}$$\Rightarrow \theta =\frac{n\pi }{3}+\frac{7\pi }{36},n\in Z$
Hence, the correct option is (C).