Find all EXACT solutions of the equation given below in the interval [0,𝜋).
tan(3x) = sqrt(3)

If there is more than one answer, enter them in a list separated by commas. Enter an exact expression; decimal approximations or symbolic trigonometric expressions such as arctan(5) will be marked incorrect.

x = 
 

To solve the equation:

\[
\tan(3x) = \sqrt{3}
\]

we first recognize that we need to find angles for which the tangent equals \(\sqrt{3}\).

### Step 1: Identify the Reference Angle
The tangent of \(\frac{\pi}{3}\) is \(\sqrt{3}\). Thus:

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

where \( n \) is an integer because the tangent function has a period of \( \pi \).

### Step 2: Solve for \( x \)
Now, we solve for \( x \) by dividing both sides by 3:

\[
x = \frac{\pi}{9} + \frac{n\pi}{3}
\]

### Step 3: Find All Solutions in the Interval \([0, \pi)\)
We need to find values of \( n \) such that \( x \) remains within \([0, \pi)\).

1. For \( n = 0 \):
   \[
   x = \frac{\pi}{9}
   \]

2. For \( n = 1 \):
   \[
   x = \frac{\pi}{9} + \frac{\pi}{3} = \frac{\pi}{9} + \frac{3\pi}{9} = \frac{4\pi}{9}
   \]

3. For \( n = 2 \):
   \[
   x = \frac{\pi}{9} + \frac{2\pi}{3} = \frac{\pi}{9} + \frac{6\pi}{9} = \frac{7\pi}{9}
   \]

4. For \( n = 3 \):
   \[
   x = \frac{\pi}{9} + \pi = \frac{\pi}{9} + \frac{9\pi}{9} = \frac{10\pi}{9}
   \]
   which is greater than \(\pi\), so it is outside the interval \([0, \pi)\).

### Final Answer
The exact solutions in the interval \([0, \pi)\) are:

\[
x = \frac{\pi}{9}, \frac{4\pi}{9}, \frac{7\pi}{9}
\]

---

Would you like further details on any of these steps?

### Related Questions
1. How would the solution change if the interval were \([0, 2\pi)\)?
2. What are the general solutions for \(\tan(3x) = -\sqrt{3}\)?
3. How would we approach this problem if we had \(\tan(3x) = 1\) instead?
4. What is the period of \(\tan(3x)\), and how does it affect the solution set?
5. Can this method be applied to solve \(\tan(kx) = c\) for any constant \( k \) and \( c \)?

#### Tip
When working with trigonometric equations, always consider the periodicity of the trigonometric function to find all solutions within a specified interval.

Find all EXACT solutions of the equation given below in the interval [0,𝜋). tan(3x) = sqrt(3)

Learn how to solve the equation tan(3x) = √3 in the interval [0, π). This step-by-step guide explains how to identify reference angles, solve for x, and find all exact solutions, including x = π/9, 4π/9, and 7π/9. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation