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

Verified step by step guidance

1. Recall that the expression cot⁻¹(\(\sqrt{3}\)) represents the angle \(\theta\) whose cotangent is \(\sqrt{3}\). In other words, \(\cot\)(\(\theta\)) = \(\sqrt{3}\).
2. Use the definition of cotangent in terms of sine and cosine: \(\cot\)(\(\theta\)) = \(\frac{\cos(\theta)}{\sin(\theta)}\). So, we need to find an angle \(\theta\) where \(\frac{\cos(\theta)}{\sin(\theta)}\) = \(\sqrt{3}\).
3. Recognize common trigonometric values: \(\cot\)(\(\theta\)) = \(\sqrt{3}\) corresponds to angles where the adjacent side is \(\sqrt{3}\) times the opposite side in a right triangle. This is a standard angle in the first quadrant.
4. Recall that \(\cot\)(\(\frac{\pi}{6}\)) = \(\sqrt{3}\) because \(\sin\)(\(\frac{\pi}{6}\)) = \(\frac{1}{2}\) and \(\cos\)(\(\frac{\pi}{6}\)) = \(\frac{\sqrt{3}\)}{2}, so \(\cot\)(\(\frac{\pi}{6}\)) = \(\frac{\frac{\sqrt{3}\)}{2}}{\(\frac{1}{2}\)} = \(\sqrt{3}\).
5. Therefore, the exact value of cot⁻¹(\(\sqrt{3}\)) is the angle \(\theta\) = \(\frac{\pi}{6}\).

Key Concepts

Inverse Cotangent Function (cot⁻¹)

The inverse cotangent function, cot⁻¹(x), returns the angle whose cotangent is x. It is the inverse of the cotangent function, which relates an angle to the ratio of the adjacent side over the opposite side in a right triangle. Understanding its range and output values is essential for finding exact angle measures.

Relationship Between Cotangent and Tangent

Cotangent is the reciprocal of tangent, meaning cot(θ) = 1/tan(θ). This relationship allows conversion between cotangent and tangent values, which can simplify finding angles using known tangent values or unit circle references.

Exact Values of Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values involving √3 and 1/√3. Recognizing these special angles helps in quickly determining the exact value of inverse trigonometric expressions without a calculator.