To find the exact value of the expression $\tan^{-1}(\sqrt{3})$, you need to determine the angle $\theta$ in radians such that:

$\tan(\theta) = \sqrt{3}$

Step-by-step Solution:

1. Recognize the trigonometric relationship: The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

2. Recall the special angle values: From trigonometric tables or the unit circle, the angle whose tangent is $\sqrt{3}$ is $\frac{\pi}{3}$ radians.

3. Verify: $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$

Thus, the exact value is:

$\boxed{\frac{\pi}{3}}$

Additional Notes:

• This value assumes the angle lies within the principal range of the $\tan^{-1}$ function, which is $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

Would you like details on verifying this or exploring tangent values further?

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Follow-up Questions:

1. What is the principal range of the $\tan^{-1}$ function?
2. How can you derive tangent values from the unit circle?
3. What is the tangent value for $\tan^{-1}(1)$?
4. What happens to $\tan^{-1}(x)$ as $x \to \infty$?
5. How do you compute $\tan^{-1}(x)$ for negative values of $x$?

Tip: Use the unit circle to memorize trigonometric values for common angles—it simplifies many calculations!