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 18°

Verified step by step guidance

1. Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). This means to find \(\csc 18^\circ\), we need to take the reciprocal of \(\sin 18^\circ\).
2. Use the given exact value for \(\sin 18^\circ\): \(\sin 18^\circ = \frac{\sqrt{5} - 1}{4}\). Substitute this into the reciprocal expression for cosecant.
3. Write the expression for \(\csc 18^\circ\) as \(\csc 18^\circ = \frac{1}{\frac{\sqrt{5} - 1}{4}}\).
4. Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\csc 18^\circ = \frac{4}{\sqrt{5} - 1}\).
5. To express the answer in a more standard exact form, rationalize the denominator by multiplying numerator and denominator by the conjugate \(\sqrt{5} + 1\), resulting in \(\csc 18^\circ = \frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)}\). Simplify the denominator using the difference of squares formula.

Key Concepts

Exact Values of Trigonometric Functions

Certain angles, like 18°, have exact trigonometric values expressible in radicals. Knowing sin 18° = (√5 - 1)/4 allows precise calculation of related functions without decimal approximations, which is essential for exact solutions in trigonometry.

Reciprocal Trigonometric Identities

The cosecant function is the reciprocal of sine, defined as csc θ = 1/sin θ. Using this identity, once sin 18° is known, csc 18° can be found exactly by taking its reciprocal, linking these functions directly.

Use of Calculator Approximations for Verification

While exact values are preferred, calculators help verify results by providing decimal approximations. Comparing the exact expression with its decimal form ensures accuracy and deepens understanding of the relationship between exact and approximate values.