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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 72
Chapter 6, Problem 72

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
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

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.

csc 72°

Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = (cot² x - 1)/(cot² x + 1)

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Textbook Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = 1 - 2 sin² x

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Textbook Question

Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

Textbook Question

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°

3
views
Textbook Question

Verify that each equation is an identity.

(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

3
views