Solve tan(3θ) = 1 / √3 for Smallest Positive θ

To solve the equation \( \tan(3\theta) = \frac{1}{\sqrt{3}} \) for the smallest positive \( \theta \), we can proceed as follows:

1. **Find the angle for \( 3\theta \):**

   We know:
   \[
   \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}}
   \]
   Thus,
   \[
   3\theta = \frac{\pi}{6} + n\pi
   \]
   where \( n \) is an integer because the tangent function is periodic with a period of \( \pi \).

2. **Solve for \( \theta \):**

   Divide both sides by 3:
   \[
   \theta = \frac{\pi}{18} + \frac{n\pi}{3}
   \]

3. **Find the smallest positive \( \theta \):**

   For the smallest positive \( \theta \), set \( n = 0 \):
   \[
   \theta = \frac{\pi}{18}
   \]

Thus, the smallest positive value of \( \theta \) is:
\[
\theta = \frac{\pi}{18}
\]

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Would you like a deeper explanation of these steps? Here are some related questions:

1. How would you solve \( \tan(4\theta) = 1/\sqrt{3} \)?
2. What is the smallest positive solution for \( \tan(2\theta) = 1 \)?
3. How does the periodicity of the tangent function affect solutions?
4. How could you find all solutions for \( \tan(3\theta) = 1/\sqrt{3} \)?
5. What if \( \tan(3\theta) = -1/\sqrt{3} \); how would the solution change?

**Tip:** When working with trigonometric equations, remember that tangent has a period of \( \pi \), meaning solutions repeat every \( \pi \) radians.

Learn how to find the smallest positive solution for the equation tan(3θ) = 1 / √3. This trigonometry problem involves understanding periodic functions, inverse trigonometric functions, and the periodicity of the tangent function. Suitable for students in Grades 10-12.

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