To evaluate the expression \( 2 \tan^{-1}\left(\frac{32}{20}\right) \), follow these steps:

### Step 1: Simplify the fraction
Simplify the fraction inside the inverse tangent function:

\[
\frac{32}{20} = \frac{8}{5} = 1.6
\]

So, the expression becomes:

\[
2 \tan^{-1}(1.6)
\]

### Step 2: Use the double-angle identity for arctan
There is a useful identity for evaluating \( 2\tan^{-1}(x) \):

\[
2\tan^{-1}(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \quad \text{if } |x| < 1
\]
or
\[
2\tan^{-1}(x) = \pi + \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \quad \text{if } x > 1
\]

Since \( x = 1.6 > 1 \), we will use the second formula:

\[
2\tan^{-1}(1.6) = \pi + \tan^{-1}\left(\frac{2(1.6)}{1 - (1.6)^2}\right)
\]

### Step 3: Calculate the expression
First, calculate \( 1.6^2 \):

\[
1.6^2 = 2.56
\]

Now substitute:

\[
2\tan^{-1}(1.6) = \pi + \tan^{-1}\left(\frac{2(1.6)}{1 - 2.56}\right)
\]

Simplify the expression inside the arctan function:

\[
\frac{2(1.6)}{1 - 2.56} = \frac{3.2}{-1.56} \approx -2.05
\]

So:

\[
2\tan^{-1}(1.6) = \pi + \tan^{-1}(-2.05)
\]

### Step 4: Approximate the value
We can now approximate the value:

\[
\tan^{-1}(-2.05) \approx -1.11 \text{ radians}
\]

Finally:

\[
2\tan^{-1}(1.6) \approx \pi - 1.11 \approx 3.14 - 1.11 = 2.03 \text{ radians}
\]

### Final Answer
\[
2\tan^{-1}\left(\frac{32}{20}\right) \approx 2.03 \text{ radians}
\]

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you find the exact value of \( \tan^{-1}(x) \) for common angles?
2. What are the other inverse trigonometric identities involving \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \)?
3. How do you evaluate an expression like \( \tan(\tan^{-1}(x)) \)?
4. What is the range of the arctan function?
5. How do you use a calculator to find the inverse tangent of a number?

**Tip:** When working with inverse trigonometric functions, remember that their outputs (angles) are usually given in radians, so it's essential to be comfortable with both degrees and radians.

Learn how to evaluate the expression 2tan^-1(32/20) step-by-step using trigonometric identities. This detailed solution covers simplifying fractions, applying the double-angle identity for arctan, and calculating the final value. Suitable for college-level trigonometry students.

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