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

검증된 단계별 안내

Recognize that the expression \( \tan^{-1} \left( \frac{\sqrt{3}}{3} \right) \) represents the angle whose tangent is \( \frac{\sqrt{3}}{3} \).

Recall the common tangent values for special angles: \( \tan 30^\circ = \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \).

Since the inverse tangent function \( \tan^{-1} \) returns the angle in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), identify the angle corresponding to \( \frac{\sqrt{3}}{3} \) within this interval.

Conclude that \( \tan^{-1} \left( \frac{\sqrt{3}}{3} \right) = \frac{\pi}{6} \) radians (or 30 degrees).

Express the exact value in radians or degrees as required by the problem.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Tangent Function (tan⁻¹ or arctan)

The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given number. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known.

Exact Values of Special Angles

Certain angles have well-known exact trigonometric values, such as tan(30°) = √3/3. Recognizing these special angles allows for finding exact values without a calculator, which is essential for solving inverse trigonometric expressions.

Relationship Between Tangent and Angles in Right Triangles

Tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Understanding this ratio helps interpret the inverse tangent function and connect numeric values to specific angles.