The general solution of `tan3x=1` is

Options

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

Answer

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

### Step 1: Identify the angle for the tangent function
We know that \( \tan(\theta) = 1 \) when \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer. Therefore, we can set up the equation:

\[ 3x = \frac{\pi}{4} + n\pi \]

### Step 2: Solve for \( x \)
Now, we need to isolate \( x \) by dividing both sides of the equation by 3:

\[ x = \frac{\frac{\pi}{4} + n\pi}{3} \]

### Step 3: Simplify the expression
We can simplify the expression further:

\[ x = \frac{\pi}{12} + \frac{n\pi}{3} \]

### Step 4: Write the general solution
The general solution for \( x \) can be expressed as:

\[ x = \frac{\pi}{12} + \frac{n\pi}{3}, \quad n \in \mathbb{Z} \]

### Final Answer
Thus, the general solution of \( \tan(3x) = 1 \) is:

\[ x = \frac{\pi}{12} + \frac{n\pi}{3}, \quad n \in \mathbb{Z} \]

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