Question

The general solution of \( \tan 3x = 1 \) is

• A. \( x = n\pi, \; n \in \mathbb{Z} \)
• B. \( x = n\left(\dfrac{\pi}{3}\right) + \dfrac{\pi}{12}, \; n \in \mathbb{Z} \)
• C. \( x = n\pi + \dfrac{\pi}{4}, \; n \in \mathbb{Z} \)
• D. \( x = n\pi \pm \dfrac{\pi}{4}, \; n \in \mathbb{Z} \)

Hint:

For equations of the form \( \tan kx = a \), always divide the general solution by \( k \).

Answer 1

The Correct Option is B

Step 1: Solve the basic equation.
\[ \tan 3x = 1 \Rightarrow 3x = \frac{\pi}{4} + n\pi \] Step 2: Solve for \( x \).
\[ x = \frac{\pi}{12} + n\frac{\pi}{3} \] Step 3: Write the general solution.
\[ x = n\left(\frac{\pi}{3}\right) + \frac{\pi}{12}, \quad n \in \mathbb{Z} \] Step 4: Conclusion.
The general solution is \( x = n\left(\dfrac{\pi}{3}\right) + \dfrac{\pi}{12} \).