The expression \(\tan^{-1}\left(\frac{132}{431}\right)\) represents the inverse tangent (or arctangent) of \(\frac{132}{431}\).

To calculate it:

\[
\theta = \tan^{-1}\left(\frac{132}{431}\right)
\]

Now, let's compute this value in radians:

\[
\theta \approx \tan^{-1}(0.3063)
\]

Using a calculator or a tool, the value of \(\tan^{-1}(0.3063)\) is approximately:

\[
\theta \approx 0.297 \text{ radians}
\]

If you prefer the answer in degrees, we can convert:

\[
\theta \approx 0.297 \times \left(\frac{180}{\pi}\right) \approx 17.01^\circ
\]

So, \(\tan^{-1}\left(\frac{132}{431}\right) \approx 0.297 \, \text{radians} \) or \(\approx 17.01^\circ \).

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Do you want further details on how the inverse tangent works or have any other questions?  
Here are five related questions you might explore:

1. How does the inverse tangent function relate to right triangles?
2. What is the derivative of \(\tan^{-1}(x)\)?
3. How can you approximate inverse trigonometric functions without a calculator?
4. What are the domain and range of the \(\tan^{-1}(x)\) function?
5. How does \(\tan^{-1}(x)\) differ from \(\cot^{-1}(x)\)?

**Tip:** Always remember that inverse trigonometric functions return angles, and their range is restricted to principal values!

The query submitted by the user: tan^{-1}(132/431)

Calculate the inverse tangent of 132/431 with detailed steps, converting the result from radians to degrees. This problem covers key trigonometric concepts and is suitable for students in Grades 10-12.

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