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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 74
Chapter 6, Problem 74

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°

Verified step by step guidance
1
Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). So, to find \(\csc 72^\circ\), we need to find \(\sin 72^\circ\) first.
Use the complementary angle identity: \(\sin 72^\circ = \cos 18^\circ\), since \(72^\circ = 90^\circ - 18^\circ\).
Express \(\cos 18^\circ\) in terms of \(\sin 18^\circ\) using the Pythagorean identity: \(\cos 18^\circ = \sqrt{1 - \sin^2 18^\circ}\).
Substitute the given exact value \(\sin 18^\circ = \frac{\sqrt{5} - 1}{4}\) into the expression for \(\cos 18^\circ\) and simplify under the square root.
Finally, calculate \(\csc 72^\circ = \frac{1}{\sin 72^\circ} = \frac{1}{\cos 18^\circ}\) using the simplified expression from the previous step.

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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 related values such as csc 72°, since 72° is complementary or related to these special angles. These exact values are foundational in advanced trigonometry.
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Reciprocal Trigonometric Functions

The cosecant function, csc θ, is the reciprocal of sin θ, defined as csc θ = 1/sin θ. To find csc 72°, one must first find sin 72° and then take its reciprocal. Understanding this relationship simplifies finding exact values when sine values are known.
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Trigonometric Identities and Angle Relationships

Using identities such as sin(90° - θ) = cos θ helps relate angles like 18° and 72°. Since 72° = 90° - 18°, sin 72° = cos 18°, which can be expressed in terms of sin 18° using Pythagorean identities. These relationships enable calculation of exact trigonometric values from known ones.
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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.

tan 72°

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Textbook Question

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

x + y = 9, 2x + y = -1

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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

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

5x - 2y + 4 = 0, 3x + 5y = 6

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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
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