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To calculate \(\tan 4\theta\) for the given triangle, we will use the result from \(\tan 2\theta\) that we previously found. The steps are as follows:

### Step 1: Recall the previously calculated \(\tan 2\theta\)
We found that:
\[
\tan 2\theta = -\frac{24}{7}
\]

### Step 2: Use the double angle identity for \(\tan 4\theta\)
The identity for the tangent of double an angle is:
\[
\tan 4\theta = \frac{2 \tan 2\theta}{1 - \tan^2 2\theta}
\]
Substitute \(\tan 2\theta = -\frac{24}{7}\) into the equation:
\[
\tan 4\theta = \frac{2 \times \left(-\frac{24}{7}\right)}{1 - \left(-\frac{24}{7}\right)^2}
\]

### Step 3: Simplify the expression
\[
\tan 4\theta = \frac{-\frac{48}{7}}{1 - \frac{576}{49}}
\]

Convert the \(1\) in the denominator to a fraction with the same denominator:
\[
\tan 4\theta = \frac{-\frac{48}{7}}{\frac{49 - 576}{49}} = \frac{-\frac{48}{7}}{-\frac{527}{49}} 
\]

Now simplify further:
\[
\tan 4\theta = \frac{-48 \times 49}{7 \times -527} = \frac{-2352}{-3689} = \frac{2352}{3689}
\]

Thus, \(\tan 4\theta = \frac{2352}{3689}\).

Would you like further details or explanations on any part of this calculation?

### Related Questions:
1. How can we calculate \(\sin 4\theta\) in this context?
2. What is the value of \(\cos 4\theta\)?
3. Can we derive a general formula for \(\tan n\theta\) for any \(n\)?
4. How would you verify \(\tan 4\theta\) using another method?
5. How does the identity for \(\tan(4\theta)\) compare to that of \(\tan(2\theta)\)?
6. What is the effect of increasing \(\theta\) on the value of \(\tan 4\theta\)?
7. Can you find the exact angle \(\theta\) using the given side lengths?
8. What are the practical applications of calculating \(\tan 4\theta\)?

### Tip:
When working with trigonometric identities involving angles like \(4\theta\) or \(2\theta\), it's crucial to carefully simplify each step to avoid errors, especially when dealing with fractions.

Learn how to calculate \( \tan 4\theta \) using the double angle identity for tangent. Follow a detailed step-by-step solution starting from \( \tan 2\theta = -\frac{24}{7} \). This problem involves advanced trigonometric concepts and is suitable for advanced high school students.

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