In Exercises 1–26, find the exact value of each expression._tan⁻¹ √3/3

Verified step by step guidance

1. Recognize that \( \tan^{-1} \) is the inverse tangent function, which means we are looking for an angle whose tangent is \( \frac{\sqrt{3}}{3} \).
2. Recall the basic angles and their tangent values. The tangent of \( 30^\circ \) or \( \frac{\pi}{6} \) radians is \( \frac{1}{\sqrt{3}} \), which simplifies to \( \frac{\sqrt{3}}{3} \).
3. Verify that \( \tan(\frac{\pi}{6}) = \frac{\sqrt{3}}{3} \) to ensure the angle is correct.
4. Conclude that the angle whose tangent is \( \frac{\sqrt{3}}{3} \) is \( \frac{\pi}{6} \) radians or \( 30^\circ \).
5. Express the final answer in the required format, either in degrees or radians, as \( \frac{\pi}{6} \) or \( 30^\circ \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹, are used to find the angle whose tangent is a given value. For example, tan⁻¹(x) returns the angle θ such that tan(θ) = x. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Tangent Function

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, is fundamental in trigonometry. Specifically, tan(θ) = sin(θ)/cos(θ). Knowing the values of common angles, such as 30°, 45°, and 60°, helps in determining the exact values of tangent and its inverse.

Special Angles in Trigonometry

Special angles, such as 30°, 45°, and 60°, have known sine, cosine, and tangent values that are often used in trigonometric calculations. For instance, tan(30°) = 1/√3 and tan(60°) = √3. Recognizing these angles allows for quick evaluation of trigonometric expressions and their inverses.