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

Verified step by step guidance

1. Recognize that \( \tan^{-1}(-\sqrt{3}) \) asks for the angle whose tangent is \(-\sqrt{3}\).
2. Recall that the tangent function is negative in the second and fourth quadrants.
3. Identify the reference angle where \( \tan(\theta) = \sqrt{3} \), which is \( \frac{\pi}{3} \) or 60 degrees.
4. Determine the angle in the second quadrant: \( \pi - \frac{\pi}{3} \).
5. Determine the angle in the fourth quadrant: \( -\frac{\pi}{3} \).

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 these functions operate is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Tangent Function and Its Values

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, can take on all real values. The specific value of tan(θ) = -√3 corresponds to angles in the second and fourth quadrants. Recognizing these angles helps in determining the exact value of the inverse tangent function.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. The first quadrant contains angles from 0 to 90 degrees, the second from 90 to 180 degrees, the third from 180 to 270 degrees, and the fourth from 270 to 360 degrees. Knowing which quadrant an angle lies in is essential for determining the sign and value of trigonometric functions.