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Ch. 5 - Trigonometric Identities
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. 5 - Trigonometric IdentitiesProblema 5.10
Capitolo 6, Problema 5.10

Find the exact value of each expression. (Do not use a calculator.)
cos(-15°)

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1
Recall the even-odd properties of the cosine function. Specifically, cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\) for any angle \(\theta\).
Apply this property to the given expression: \(\cos(-15^\circ) = \cos(15^\circ)\).
Express \(15^\circ\) as a difference of two common angles whose cosine and sine values are known, for example, \(15^\circ = 45^\circ - 30^\circ\).
Use the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). Substitute \(a = 45^\circ\) and \(b = 30^\circ\) to get \(\cos(15^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ\).
Recall the exact values: \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\). Substitute these into the expression to write the exact value of \(\cos(-15^\circ)\).

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Even-Odd Properties of Trigonometric Functions

Cosine is an even function, meaning cos(-θ) = cos(θ). This property allows us to simplify expressions with negative angles by converting them to positive angles without changing the value.
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Even and Odd Identities

Angle Addition and Subtraction Formulas

These formulas express trigonometric functions of sums or differences of angles, such as cos(a - b) = cos a cos b + sin a sin b. They are essential for finding exact values of angles not commonly found on the unit circle.
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Adding and Subtracting Complex Numbers

Exact Values of Special Angles

Certain angles like 0°, 30°, 45°, 60°, and 90° have known exact sine and cosine values. Using these, along with angle formulas, helps compute exact values for other angles like 15° by expressing them as sums or differences of special angles.
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45-45-90 Triangles
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