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The problem asks you to estimate all solutions of the equation \( \tan(x) = 2.9 \) in the interval \( 0 \leq x \leq 2\pi \), using a graph and rounding the answers to three decimal places.

### Approach:
To solve this, you can use the fact that the tangent function is periodic with a period of \( \pi \). This means the function will repeat its values at intervals of \( \pi \). The general solution for \( \tan(x) = 2.9 \) can be found by graphing \( y = \tan(x) \) and identifying the points where the graph intersects the horizontal line \( y = 2.9 \).

1. **First solution**: The first intersection occurs in the first period, between \( 0 \) and \( \pi \).
2. **Second solution**: Since the tangent function is periodic, there will be a second solution between \( \pi \) and \( 2\pi \).

### Steps:
- Using a calculator or graphing tool, graph \( y = \tan(x) \) and identify the points where \( y = 2.9 \).
- Find these \( x \)-coordinates in radians, and round the values to three decimal places.

### Results (Graphical Estimation):
From a graph of \( y = \tan(x) \):
- **First solution**: The first value of \( x \) is approximately **1.28** radians.
- **Second solution**: The second value of \( x \) is approximately **4.42** radians.

Therefore, the two solutions to \( \tan(x) = 2.9 \) in the interval \( [0, 2\pi] \) are:

\[
x_1 \approx 1.281 \quad \text{and} \quad x_2 \approx 4.423
\]

Let me know if you'd like more details or clarifications.

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Here are 5 follow-up questions you could explore:
1. What is the general solution for \( \tan(x) = 2.9 \) considering the periodic nature of the tangent function?
2. How does the period of the tangent function affect the number of solutions in a given interval?
3. Can you explain why the tangent function has vertical asymptotes?
4. How would the solutions change if the equation were \( \tan(x) = -2.9 \)?
5. What are the key differences between the sine, cosine, and tangent functions in terms of periodicity?

### Tip:
When working with trigonometric functions, always pay attention to their periodicity, as this will determine how many solutions exist within a given interval.

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

Learn how to estimate the solutions of the trigonometric equation tan(x) = 2.9 within the interval 0 to 2π, using a graph. This example explains the periodic nature of the tangent function and the importance of rounding solutions to three decimal places.

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