The principal solutions of $\tan 3 \theta=-1$ are

The principal solutions of $\tan 3 \theta=-1$ are
  1. $\left\{\frac{\pi}{4}, \frac{7 \pi}{12}, \frac{11 \pi}{12}, \frac{5 \pi}{4}, \frac{19 \pi}{12}, \frac{23 \pi}{12}\right\}$
  2. $\left\{\frac{\pi}{4}, \frac{7 \pi}{12}, \frac{11 \pi}{12}, \frac{\pi}{16}, \frac{19 \pi}{12}, \frac{23 \pi}{24}\right\}$
  3. $\left\{\frac{\pi}{4}, \frac{\pi}{12}\right\}$
  4. $\left\{\frac{\pi}{4}, \frac{\pi}{12}, \frac{13 \pi}{12}, \frac{7 \pi}{4}, \frac{19 \pi}{4}, \frac{23 \pi}{12}\right\}$

Solution

$\begin{aligned} & \tan 3 \theta=-1=\tan \frac{3 \pi}{4} \\ & \Rightarrow 3 \theta=n \pi \pm \frac{3 \pi}{4} \\ & \Rightarrow \theta=\frac{n \pi}{3} \pm \frac{\pi}{4} \\ & \Rightarrow \theta \in\left\{\frac{\pi}{4}, \frac{7 \pi}{12}, \frac{11 \pi}{12}, \frac{15 \pi}{12}, \frac{19 \pi}{12}, \frac{23 \pi}{12}\right\}\end{aligned}$

Asked in: MHT CET 2022 (07 Aug Shift 1)

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