We need to solve the equation:

$14 \tan(\pi \theta - 4) + 31 = 10$

Steps:

1. Isolate the trigonometric term: $14 \tan(\pi \theta - 4) = 10 - 31$ $14 \tan(\pi \theta - 4) = -21$ $\tan(\pi \theta - 4) = \frac{-21}{14}$ $\tan(\pi \theta - 4) = -\frac{3}{2}$

2. Solve for $\pi \theta - 4$: Recall that $\tan(x) = y$ implies: $x = \tan^{-1}(y) + n\pi, \quad n \in \mathbb{Z}$ Applying this to our equation: $\pi \theta - 4 = \tan^{-1}\left(-\frac{3}{2}\right) + n\pi, \quad n \in \mathbb{Z}$

3. Calculate $\tan^{-1}(-3/2)$: Using approximate values or a calculator: $\tan^{-1}\left(-\frac{3}{2}\right) \approx -0.9828 \, \text{radians}.$

   Substituting: $\pi \theta - 4 = -0.9828 + n\pi$

4. Solve for $\theta$: $\pi \theta = 4 - 0.9828 + n\pi$ $\pi \theta = 3.0172 + n\pi$ Divide through by $\pi$: $\theta = \frac{3.0172}{\pi} + n$

   Approximating $\pi \approx 3.1416$: $\theta = \frac{3.0172}{3.1416} + n$ $\theta \approx 0.9602 + n$

General Solution:

$\theta = 0.9602 + n, \quad n \in \mathbb{Z}.$

Would you like a more detailed step-by-step breakdown or a specific evaluation for $n$?

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Follow-up Questions:

1. What values of $n$ are most relevant to your problem?
2. Should the solution be restricted to a specific interval, such as $[0, 1]$?
3. Would you like the graph of this function to visualize the solutions?
4. Are there additional trigonometric equations you'd like solved?
5. Do you want to verify this solution with numerical methods?

Tip:

When working with trigonometric equations, always ensure you consider the periodicity of the functions to capture all possible solutions!