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°

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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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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.

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.

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.

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°