Question

General solution of $\tan 5\theta = \cot 2\theta$ is:

• A. \( \theta = \frac{n\pi}{7} + \frac{\pi}{14} \)
• B. \( \theta = \frac{n\pi}{7} + \frac{\pi}{5} \)
• C. \( \theta = \frac{n\pi}{7} + \frac{\pi}{2} \)
• D. \( \theta = \frac{n\pi}{7} + \frac{\pi}{3} \)

Hint:

When resolving matching expressions like \(\tan A = \cot B\), you can bypass writing out the intermediate steps by jumping straight to a handy universal shortcut identity: \( A + B = n\pi + \frac{\pi}{2} \). Substituting our specific parameters directly gives \( 5\theta + 2\theta = n\pi + \frac{\pi}{2} \implies 7\theta = n\pi + \frac{\pi}{2} \), matching our core result in under 5 seconds!

Answer 1

The Correct Option is A

Concept: To find the general solution of a trigonometric equation involving a mixed pairing like \(\tan A = \cot B\), we first transform both sides into matching trigonometric functions using complementary angle properties. The standard transformations and theorems used are:
• Complementary Angle Identity: \[ \cot \alpha = \tan\left(\frac{\pi}{2} - \alpha\right) \]
• General Solution Theorem for Tangent: If \(\tan \phi = \tan \alpha\), then the general solution is uniquely defined as: \[ \phi = n\pi + \alpha, \quad \text{where } n \in \mathbb{Z} \]

Step 1: Converting the equation to matching tangent terms.
Given the expression: \[ \tan 5\theta = \cot 2\theta \] Using the complementary conversion rule, rewrite the right-hand side in terms of tangent: \[ \cot 2\theta = \tan\left(\frac{\pi}{2} - 2\theta\right) \] Substitute this back into the core equation system: \[ \tan 5\theta = \tan\left(\frac{\pi}{2} - 2\theta\right) \]

Step 2: Applying the general solution theorem to isolate the variable.
Equating the arguments through our general tangent theorem: \[ 5\theta = n\pi + \left(\frac{\pi}{2} - 2\theta\right) \] Now, rearrange the terms to group all \(\theta\) parameters together on the left-hand side: \[ 5\theta + 2\theta = n\pi + \frac{\pi}{2} \] \[ 7\theta = n\pi + \frac{\pi}{2} \]

Step 3: Solving for $\theta$.
Dividing both sides cleanly by 7 gives our final isolated general solution expression: \[ \theta = \frac{n\pi}{7} + \frac{\pi}{2 \times 7} \] \[ \theta = \frac{n\pi}{7} + \frac{\pi}{14} \]