Step 1: Understanding the Concept:
We need to solve a general trigonometric equation.
The best approach is to convert both sides to the same trigonometric function, either both tangent or both cotangent.
Step 2: Key Formula or Approach:
Complementary angle identity: $\cot \theta = \tan\left(\frac{\pi}{2} - \theta\right)$.
General solution formula for $\tan x = \tan y$:
If $\tan x = \tan y$, then $x = n\pi + y$, where $n$ is any integer ($n \in \mathbb{Z}$).
Step 3: Detailed Explanation:
Given equation:
\[ \tan 3\theta = \cot \theta \]
Convert $\cot \theta$ to $\tan$ using the identity:
\[ \tan 3\theta = \tan\left(\frac{\pi}{2} - \theta\right) \]
Now we have an equation of the form $\tan A = \tan B$.
Applying the general solution formula:
\[ 3\theta = n\pi + \left(\frac{\pi}{2} - \theta\right) \]
where $n \in \mathbb{Z}$.
Rearrange to solve for $\theta$. Bring $-\theta$ to the left side:
\[ 3\theta + \theta = n\pi + \frac{\pi}{2} \]
\[ 4\theta = \frac{2n\pi + \pi}{2} \]
Factor out $\pi$ in the numerator:
\[ 4\theta = \frac{(2n + 1)\pi}{2} \]
Divide by 4 to isolate $\theta$:
\[ \theta = \frac{(2n + 1)\pi}{2 \times 4} \]
\[ \theta = \frac{(2n + 1)\pi}{8} \]
where $n$ is an integer.
Step 4: Final Answer:
The general solution is $\theta = \frac{(2n+1)\pi}{8}, n \in \mathbb{Z}$.