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The problem provided is: 

\[
\text{If } \tan(3\theta) = \frac{1}{\sqrt{3}}, \text{ what is the smallest positive value of } \theta?
\]

To solve for \(\theta\), let's break it down step by step:

1. Recall that \(\tan(3\theta) = \frac{1}{\sqrt{3}}\) is equivalent to \(\tan(30^\circ)\) or \(\tan\left(\frac{\pi}{6}\right)\), because \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

2. Therefore, we set:
\[
3\theta = \frac{\pi}{6}
\]

3. Solving for \(\theta\):
\[
\theta = \frac{\pi}{6} \div 3 = \frac{\pi}{18}
\]

Thus, the smallest positive value of \(\theta\) is \(\frac{\pi}{18}\), which corresponds to **Response 1** in the options.

Would you like a further breakdown or have any questions? Here are five related questions for practice:

1. What is the general solution for \(\theta\) if \(\tan(3\theta) = \frac{1}{\sqrt{3}}\)?
2. How do you find the principal solution of a trigonometric equation?
3. What is the period of the tangent function?
4. How does multiplying the angle inside a trigonometric function affect the period of the function?
5. How would the solution change if the equation was \(\tan(2\theta) = \frac{1}{\sqrt{3}}\)?

**Tip:** Always check the reference angle and the quadrant when solving trigonometric equations to ensure you find all possible solutions.

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

Learn how to solve the trigonometric equation tan(3θ) = 1/√3 step by step to find the smallest positive value of θ. This problem focuses on trigonometry concepts involving tangent functions and angle relationships. Suitable for Grades 10-12.

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