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°

검증된 단계별 안내

Recall the complementary angle relationship between sine and cosine: \(\sin(18^\circ) = \cos(72^\circ)\) because \(72^\circ = 90^\circ - 18^\circ\).

Use the given exact value \(\sin(18^\circ) = \frac{\sqrt{5} - 1}{4}\) to write \(\cos(72^\circ) = \frac{\sqrt{5} - 1}{4}\).

Express \(\tan(72^\circ)\) in terms of sine and cosine: \(\tan(72^\circ) = \frac{\sin(72^\circ)}{\cos(72^\circ)}\).

Use the Pythagorean identity to find \(\sin(72^\circ)\): \(\sin(72^\circ) = \sqrt{1 - \cos^2(72^\circ)}\). Substitute \(\cos(72^\circ) = \frac{\sqrt{5} - 1}{4}\) into this expression.

Finally, substitute the values of \(\sin(72^\circ)\) and \(\cos(72^\circ)\) into the tangent expression to get the exact value of \(\tan(72^\circ)\).

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Exact Values of Special Angles

Certain angles like 18°, 36°, 54°, and 72° have exact trigonometric values expressible using radicals. Knowing sin 18° = (√5 - 1)/4 allows derivation of other values through identities, avoiding decimal approximations and enabling precise calculations.

Trigonometric Identities

Identities such as tan θ = sin θ / cos θ and angle relationships like 72° = 90° - 18° help transform known values into unknown ones. Using these identities, one can express tan 72° in terms of sin 18° and related functions for exact evaluation.

Calculator Approximations for Verification

While exact values are expressed symbolically, calculators provide decimal approximations to verify results. Comparing the exact expression's decimal form with calculator output ensures correctness and deepens understanding of the relationship between symbolic and numeric values.

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°