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°

검증된 단계별 안내

Recall the given exact value: \(\sin 18^\circ = \frac{\sqrt{5} - 1}{4}\).

Use the identity for sine of supplementary angles: \(\sin(180^\circ - \theta) = \sin \theta\). Since \(162^\circ = 180^\circ - 18^\circ\), we have \(\sin 162^\circ = \sin 18^\circ\).

Substitute the known exact value into the expression: \(\sin 162^\circ = \frac{\sqrt{5} - 1}{4}\).

Optionally, verify the result by calculating the decimal approximation of \(\frac{\sqrt{5} - 1}{4}\) and comparing it with a calculator value of \(\sin 162^\circ\).

Summarize that the exact value of \(\sin 162^\circ\) is the same as \(\sin 18^\circ\) due to the supplementary angle identity.

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주요 개념

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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.

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°