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.
sin 162°

Verified step by step guidance

1. Recall the given exact value: \(\sin 18^\circ = \frac{\sqrt{5} - 1}{4}\).
2. Use the identity for sine of supplementary angles: \(\sin(180^\circ - \theta) = \sin \theta\). Since \(162^\circ = 180^\circ - 18^\circ\), we have \(\sin 162^\circ = \sin 18^\circ\).
3. Substitute the known exact value into the expression: \(\sin 162^\circ = \frac{\sqrt{5} - 1}{4}\).
4. Optionally, verify the result by calculating the decimal approximation of \(\frac{\sqrt{5} - 1}{4}\) and comparing it with a calculator value of \(\sin 162^\circ\).
5. Summarize that the exact value of \(\sin 162^\circ\) is the same as \(\sin 18^\circ\) due to the supplementary angle identity.

Key Concepts

Reference Angles and Angle Relationships

Understanding how angles relate to each other on the unit circle is essential. For example, 162° is in the second quadrant and can be expressed as 180° - 18°, which helps use known values like sin 18° to find sin 162° using symmetry properties.

Sine Function Properties and Identities

The sine function has specific properties such as sin(180° - θ) = sin θ. This identity allows the exact value of sin 162° to be found directly from sin 18°, simplifying calculations and avoiding approximation errors.

Exact Values and Surd Expressions in Trigonometry

Certain angles have exact trigonometric values expressed in surds, like sin 18° = (√5 - 1)/4. Recognizing and manipulating these exact forms is crucial for precise answers rather than decimal approximations, especially in advanced trigonometry.