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We are given the equation:

\[
\tan(3\theta) = \frac{1}{\sqrt{3}}
\]

We are tasked with finding the smallest positive value of \(\theta\).

### Step 1: Recognizing the tangent value
We know from trigonometric values that:

\[
\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}
\]

Thus, we can equate:

\[
3\theta = \frac{\pi}{6}
\]

### Step 2: General solution for tangent
Since the tangent function is periodic with a period of \(\pi\), the general solution for \(\tan(3\theta) = \frac{1}{\sqrt{3}}\) is:

\[
3\theta = \frac{\pi}{6} + n\pi \quad \text{where} \, n \, \text{is an integer}.
\]

### Step 3: Solve for \(\theta\)
Now, solving for \(\theta\):

\[
\theta = \frac{\pi}{18} + \frac{n\pi}{3}
\]

### Step 4: Finding the smallest positive \(\theta\)
For \(n = 0\), we get:

\[
\theta = \frac{\pi}{18}
\]

Thus, the smallest positive value of \(\theta\) is:

\[
\theta = \frac{\pi}{18}
\]

### Final Answer:
The smallest positive value of \(\theta\) is \(\frac{\pi}{18}\).

Do you want more details or have any further questions on this solution?

### Related Questions:
1. How do you find general solutions to trigonometric equations like \(\tan(x)\)?
2. What are the key values of \(\tan(x)\) in the unit circle?
3. How does the period of \(\tan(x)\) affect the general solution?
4. Can this method be applied to equations involving \(\sin(x)\) or \(\cos(x)\)?
5. What is the significance of adding multiples of \(\pi\) in the solution?

### Tip:
For equations involving \(\tan(k\theta)\), always remember to account for the periodic nature of tangent by adding multiples of \(\pi\) to your solution.

If tan(3θ) = 1 / √3, what is the smallest positive value of θ?

Learn how to solve the equation tan(3θ) = 1 / √3 and find the smallest positive value of θ. This problem explores trigonometric concepts like the periodic nature of the tangent function and general solutions. Suitable for Grades 10-12.

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