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. Recall the given exact value: \(\sin 18^\circ = \frac{\sqrt{5} - 1}{4}\).
2. Use the Pythagorean identity to find \(\cos 18^\circ\): \(\cos 18^\circ = \sqrt{1 - \sin^2 18^\circ}\).
3. Substitute the value of \(\sin 18^\circ\) into the expression for \(\cos 18^\circ\) and simplify under the square root.
4. Recall that \(\cot 18^\circ = \frac{\cos 18^\circ}{\sin 18^\circ}\), so write \(\cot 18^\circ\) as a fraction using the values found.
5. Simplify the fraction to express \(\cot 18^\circ\) in exact form, possibly rationalizing the denominator if needed.

Key Concepts

Exact Values of Special Angles

Certain angles like 18°, 30°, 45°, 60°, and 90° have known exact trigonometric values often expressed using square roots. For example, sin 18° = (√5 - 1)/4 is a derived exact value. Understanding these helps in calculating other trigonometric functions without relying solely on approximations.

Trigonometric Identities

Identities such as cot θ = cos θ / sin θ or cot θ = 1 / tan θ allow conversion between trigonometric functions. Using the given sin 18°, one can find cot 18° by first determining cos 18° through identities like cos² θ + sin² θ = 1, then applying the cotangent definition.

Using Algebraic Manipulation to Derive Exact Values

Finding exact trigonometric values often requires algebraic manipulation of expressions involving square roots and fractions. This includes simplifying radicals and rationalizing denominators to express results in simplest exact form, which is essential for precise answers beyond decimal approximations.