Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
cot 18°

Verified step by step guidance

1. Start by recalling the identity for cotangent: \( \cot \theta = \frac{1}{\tan \theta} \).
2. Use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) to express \( \cot 18^\circ \) as \( \frac{\cos 18^\circ}{\sin 18^\circ} \).
3. Since \( \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \), use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos 18^\circ \).
4. Substitute \( \sin 18^\circ \) into the Pythagorean identity to solve for \( \cos 18^\circ \).
5. Finally, substitute the values of \( \cos 18^\circ \) and \( \sin 18^\circ \) into \( \cot 18^\circ = \frac{\cos 18^\circ}{\sin 18^\circ} \) to find the exact value.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. Understanding these identities is essential for manipulating and simplifying trigonometric expressions, such as finding cotangent values from sine or cosine.

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function and can be expressed as cot(θ) = cos(θ)/sin(θ). For specific angles, such as 18°, knowing the sine and cosine values allows for the direct calculation of cotangent. This relationship is crucial for solving problems that require finding cotangent values using sine or cosine.

Exact Values in Trigonometry

Exact values in trigonometry refer to specific trigonometric function values that can be expressed in terms of radicals or fractions rather than decimals. For example, sin(18°) = (√5 - 1)/4 is an exact value. Using known exact values and identities enables the calculation of other trigonometric functions without relying on approximations, which is particularly useful in advanced trigonometric problems.