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

Step 1 : Understanding the Question:
This problem asks us to find the general solution for $\theta$ in the trigonometric equation $\tan 5\theta = \cot 2\theta$. The equation contains two different trigonometric functions, tangent and cotangent, acting on different multiples of $\theta$. To find the solution, we need to convert both sides of the equation into the same trigonometric function and then apply the standard general solution formula.
Step 2 : Key Formulas and Approach:
We can convert the cotangent function to the tangent function using the complementary angle identity:
\[ \cot \phi = \tan\left(\frac{\pi}{2} - \phi\right) \]
This conversion allows us to write the equation in the standard form:
\[ \tan A = \tan B \]
The general solution for this standard trigonometric equation is given by:
\[ A = n\pi + B, \quad \text{where } n \in \mathbb{Z} \]
We will apply this general formula to our transformed equation and isolate the variable $\theta$.
Step 3 : Detailed Explanation:

We start with the given trigonometric equation: $\tan 5\theta = \cot 2\theta$.

To express both sides in terms of the tangent function, we apply the complementary angle identity to the right-hand side: $\cot 2\theta = \tan\left(\frac{\pi}{2} - 2\theta\right)$.

Substituting this back into the original equation gives: $\tan 5\theta = \tan\left(\frac{\pi}{2} - 2\theta\right)$.

Next, we apply the general solution formula for the tangent function, which leads to the equation: $5\theta = n\pi + \left(\frac{\pi}{2} - 2\theta\right)$, where $n$ is any integer.

To solve for $\theta$, we first gather all terms containing $\theta$ on the left-hand side of the equation. We add $2\theta$ to both sides: $5\theta + 2\theta = n\pi + \frac{\pi}{2}$.

Simplifying the left-hand side yields: $7\theta = n\pi + \frac{\pi}{2}$.

Finally, we isolate $\theta$ by dividing the entire equation by 7: $\theta = \frac{n\pi}{7} + \frac{\pi}{2 \times 7}$.

This simplifies to the general solution: $\theta = \frac{n\pi}{7} + \frac{\pi}{14}$.

Step 4 : Final Answer:
The general solution of the equation is $\theta = \frac{n\pi}{7} + \frac{\pi}{14}$, which corresponds to Option (A).