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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 52
Capitolo 3, Problema 52

Give the exact value of each expression. See Example 5. cos 30°

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1
Recall that 30° is one of the special angles in trigonometry, and its cosine value is well-known from the unit circle or special right triangles.
Recognize that the cosine of 30° corresponds to the adjacent side over the hypotenuse in a 30°-60°-90° right triangle.
In a 30°-60°-90° triangle, the sides are in the ratio 1 (opposite 30°) : \(\sqrt{3}\) (adjacent 30°) : 2 (hypotenuse).
Therefore, the cosine of 30° is the length of the adjacent side over the hypotenuse, which can be written as \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
Write down the exact value using the simplified radical form without decimal approximation.

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Unit Circle and Special Angles

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Special angles like 30°, 45°, and 60° have well-known sine and cosine values derived from equilateral and right triangles, which help in finding exact trigonometric values.
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Introduction to the Unit Circle

Cosine Function Definition

Cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. On the unit circle, cosine corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle, allowing exact values to be determined for standard angles.
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Graph of Sine and Cosine Function

Exact Values of Cosine for 30°

The exact value of cos 30° is derived from the 30°-60°-90° triangle, where the sides are in the ratio 1:√3:2. Cos 30° equals √3/2, representing the adjacent side over the hypotenuse, providing a precise, simplified radical form rather than a decimal approximation.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°
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