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

Verified step by step guidance

1. Recognize that \( \cot^{-1}(x) \) is the inverse cotangent function, which gives the angle whose cotangent is \( x \).
2. Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, if \( \cot(\theta) = \sqrt{3} \), then \( \tan(\theta) = \frac{1}{\sqrt{3}} \).
3. Identify the angle \( \theta \) for which \( \tan(\theta) = \frac{1}{\sqrt{3}} \). This is a common angle in trigonometry.
4. Recall that \( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \). Therefore, \( \theta = \frac{\pi}{6} \).
5. Conclude that \( \cot^{-1}(\sqrt{3}) = \frac{\pi}{6} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cot⁻¹, are used to find angles when given a trigonometric ratio. For example, cot⁻¹(x) gives the angle whose cotangent is x. Understanding these functions is crucial for solving problems that require angle determination from known ratios.

Cotangent Function

The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle, or as the reciprocal of the tangent function. Specifically, cot(θ) = 1/tan(θ). Knowing the values of cotangent for common angles helps in finding exact values for inverse cotangent expressions.

Special Angles in Trigonometry

Special angles, such as 30°, 45°, and 60°, have known trigonometric values that are often used in calculations. For instance, cot(30°) = √3 and cot(60°) = 1/√3. Recognizing these angles and their corresponding values is essential for quickly determining the exact values of trigonometric expressions.