General solution of `tan((2x)/(3))=sqrt(3)` is

Options

A. `(2n+1)(pi)/(3), ninZ`
B. `(3n+1)(pi)/(3), ninZ`
C. `(2n+1)(pi)/(2), ninZ`
D. `(3n+1)(pi)/(2), ninZ`

Answer

To find the general solution of the equation \( \tan\left(\frac{2x}{3}\right) = \sqrt{3} \), we can follow these steps:

### Step 1: Identify the angle where tangent equals \(\sqrt{3}\)
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus, we can set:
\[ \frac{2x}{3} = \frac{\pi}{3} + n\pi \]
where \( n \) is any integer. This accounts for the periodic nature of the tangent function.

### Step 2: Solve for \( x \)
To isolate \( x \), we first multiply both sides of the equation by 3:
\[ 2x = \pi + 3n\pi \]
Now, divide by 2:
\[ x = \frac{\pi}{2} + \frac{3n\pi}{2} \]

### Step 3: Simplify the expression
We can factor out \(\frac{\pi}{2}\):
\[ x = \frac{\pi}{2}(1 + 3n) \]
where \( n \) is any integer.

### Final Answer
Thus, the general solution for the equation \( \tan\left(\frac{2x}{3}\right) = \sqrt{3} \) is:
\[ x = \frac{\pi}{2}(1 + 3n), \quad n \in \mathbb{Z} \]

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