https://file.aiquickdraw.com/mathbot/file/1729203055911wkb12in4.jpg

The problem asks for all solutions of the equation \( \tan(x) = 2.9 \) between \( 0 \) and \( 2\pi \), with answers rounded to three decimal places.

Here's how we can approach this:

### Step-by-step process:

1. **Understanding the tangent function**: 
   The tangent function is periodic with a period of \( \pi \). This means that if \( x_1 \) is a solution, then any other solution can be found by adding or subtracting multiples of \( \pi \). Therefore, in the interval \( [0, 2\pi] \), there will be two solutions, one in the first period \( [0, \pi] \) and one in the second period \( [\pi, 2\pi] \).

2. **Finding the first solution**:
   Using a calculator or graphing tool, we can approximate the value of \( x \) for \( \tan(x) = 2.9 \). The principal solution (the one in \( [0, \pi] \)) is approximately:
   \[
   x_1 \approx 1.277
   \]
   This value was already provided in the image you uploaded.

3. **Finding the second solution**:
   Since \( \tan(x) \) repeats every \( \pi \), the second solution can be found by adding \( \pi \) to the first solution:
   \[
   x_2 = x_1 + \pi \approx 1.277 + 3.141 = 4.418
   \]

4. **Rounding the answers**:
   The two solutions, rounded to three decimal places, are:
   \[
   x_1 \approx 1.277 \quad \text{and} \quad x_2 \approx 4.418
   \]

### Correction to your inputs:
You entered 1.277 (which is correct) but the second value 4.681 is incorrect. The correct second value should be approximately **4.418**.

Would you like to go over any of the steps in more detail, or clarify any questions?

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Here are 5 related questions to deepen your understanding:

1. Why does the tangent function repeat every \( \pi \) instead of \( 2\pi \)?
2. How would the solutions change if the interval was between \( 0 \) and \( 4\pi \)?
3. What happens if the equation were \( \tan(x) = -2.9 \) instead?
4. How can you use the unit circle to estimate solutions for \( \tan(x) \)?
5. How would you solve this equation algebraically if the tangent function were involved in a more complex equation?

**Tip**: Always remember to account for the periodicity of trigonometric functions when solving equations involving them in bounded intervals!

Use a graph to estimate all the solutions of tan(x) = 2.9 between 0 and 2π, and round your answers to three decimal places.

Learn how to solve the equation tan(x) = 2.9 between 0 and 2π, with step-by-step guidance. This problem involves trigonometric concepts, focusing on the periodic nature of the tangent function. Suitable for high school students.

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