Give the exact value of each expression. See Example 5. cot 45°

Verified step by step guidance

1. Recall the definition of cotangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
2. Substitute \(\theta = 45^\circ\) into the formula: \(\cot 45^\circ = \frac{\cos 45^\circ}{\sin 45^\circ}\).
3. Use the known exact values for sine and cosine at \(45^\circ\): \(\sin 45^\circ = \frac{\sqrt{2}}{2}\) and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
4. Replace the sine and cosine values in the expression: \(\cot 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\).
5. Simplify the fraction by dividing the numerator by the denominator to find the exact value of \(\cot 45^\circ\).

Key Concepts

Definition of Cotangent

Cotangent is a trigonometric function defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, cot(θ) = 1 / tan(θ). It represents the reciprocal of the tangent function and is useful for finding exact values of angles.

Exact Values of Common Angles

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values. For 45°, the cotangent value can be derived from the properties of an isosceles right triangle, where the legs are equal, leading to cot 45° = 1.

Using Reciprocal Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as cot(θ) = 1 / tan(θ). This identity allows calculation of cotangent values by first knowing or finding the tangent value, simplifying the process of finding exact trigonometric values.