The general solution of `"tan" 3x =1`, is

Options

A. `n pi + (pi)/(4)`
B. `(npi)/(3) + (pi)/(12)`
C. `n pi`
D. `n pi +- (pi)/(4)`

Answer

To find the general solution of the equation \( \tan(3x) = 1 \), we can follow these steps:

### Step 1: Understand the basic property of the tangent function
The equation \( \tan(A) = \tan(B) \) implies that:
\[ A = B + n\pi \quad \text{for } n \in \mathbb{Z} \]
This means that if two angles have the same tangent value, they differ by an integer multiple of \( \pi \).

### Step 2: Set up the equation
Given \( \tan(3x) = 1 \), we can express this as:
\[ 3x = \frac{\pi}{4} + n\pi \]
Here, we use \( \frac{\pi}{4} \) because \( \tan\left(\frac{\pi}{4}\right) = 1 \).

### Step 3: Solve for \( x \)
Now, we need to isolate \( x \) in the equation:
\[ 3x = \frac{\pi}{4} + n\pi \]
Dividing both sides by 3 gives:
\[ x = \frac{\frac{\pi}{4} + n\pi}{3} \]
This can be simplified to:
\[ x = \frac{\pi}{12} + \frac{n\pi}{3} \]

### Step 4: Write the general solution
Thus, the general solution for the equation \( \tan(3x) = 1 \) is:
\[ x = \frac{\pi}{12} + \frac{n\pi}{3}, \quad n \in \mathbb{Z} \]

### Conclusion
The general solution of \( \tan(3x) = 1 \) is:
\[ x = \frac{n\pi}{3} + \frac{\pi}{12} \]