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.
tan 72°

Verified step by step guidance

1. Recognize that \( \tan 72^\circ \) can be expressed using the identity \( \tan(90^\circ - \theta) = \cot \theta \). Therefore, \( \tan 72^\circ = \cot 18^\circ \).
2. Use the identity \( \cot \theta = \frac{1}{\tan \theta} \) to express \( \cot 18^\circ \) in terms of \( \tan 18^\circ \).
3. Recall the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and use the given \( \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \) to find \( \cos 18^\circ \) using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
4. Calculate \( \cos 18^\circ \) by substituting \( \sin 18^\circ \) into the Pythagorean identity and solving for \( \cos 18^\circ \).
5. Substitute \( \sin 18^\circ \) and \( \cos 18^\circ \) into \( \tan 18^\circ = \frac{\sin 18^\circ}{\cos 18^\circ} \) to find \( \tan 18^\circ \), and then use \( \cot 18^\circ = \frac{1}{\tan 18^\circ} \) to find \( \tan 72^\circ \).

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 identities, angle sum and difference identities, and double angle formulas. These identities are essential for simplifying expressions and solving trigonometric equations, such as finding the value of tan 72° using known values like sin 18°.

Angle Relationships

Understanding angle relationships in trigonometry is crucial for solving problems involving complementary and supplementary angles. For instance, tan(90° - θ) = cot(θ) and tan(180° - θ) = -tan(θ). In this case, recognizing that tan 72° can be expressed in terms of sin 18° through complementary angles aids in finding the exact value using known trigonometric values.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific angles where the sine, cosine, and tangent can be expressed as simple fractions or radicals rather than decimal approximations. For example, sin 18° = (√5 - 1)/4 is an exact value. Knowing these exact values allows for precise calculations and the ability to derive other trigonometric values, such as tan 72°, through identities and relationships.