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Ch. 1 - Trigonometric Functions
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. 1 - Trigonometric FunctionsProblema 98
Capitolo 2, Problema 98

Use trigonometric function values of quadrantal angles to evaluate each expression. [cos(―180°)]² + [sin(― 180°)]²

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Recall the Pythagorean identity in trigonometry: \(\cos^2 \theta + \sin^2 \theta = 1\) for any angle \(\theta\).
Identify the angle given in the problem: \(-180^\circ\). Note that \(-180^\circ\) is a quadrantal angle, which means it lies on the x-axis or y-axis of the unit circle.
Evaluate \(\cos(-180^\circ)\) and \(\sin(-180^\circ)\) using the unit circle values for quadrantal angles. Remember that \(\cos(-\theta) = \cos \theta\) and \(\sin(-\theta) = -\sin \theta\).
Square the values obtained for \(\cos(-180^\circ)\) and \(\sin(-180^\circ)\) separately, resulting in \([\cos(-180^\circ)]^2\) and \([\sin(-180^\circ)]^2\).
Add the squared values together as per the expression: \([\cos(-180^\circ)]^2 + [\sin(-180^\circ)]^2\). Use the Pythagorean identity to confirm the result.

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Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the coordinate plane, typically 0°, 90°, 180°, 270°, and 360°. Their sine and cosine values are either 0, 1, or -1, which simplifies calculations. Understanding these values helps evaluate trigonometric expressions involving such angles.
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Quadratic Formula

Trigonometric Function Values at Specific Angles

The sine and cosine of specific angles, especially quadrantal angles, have well-known exact values. For example, cos(180°) = -1 and sin(180°) = 0. Knowing these values allows direct substitution into expressions without approximation.
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Introduction to Trigonometric Functions

Pythagorean Identity

The Pythagorean identity states that for any angle θ, (cos θ)² + (sin θ)² = 1. This fundamental identity confirms that the sum of the squares of sine and cosine of the same angle always equals one, which is key to evaluating the given expression.
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