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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 103
Capitolo 2, Problema 103

Concept Check Suppose that ―90° < θ < 90° .   Find the sign of each function value. cos(θ―180°)

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Recall the given range for \( \theta \): \( -90^\circ < \theta < 90^\circ \). This means \( \theta \) is in either Quadrant I or Quadrant IV.
Rewrite the expression inside the cosine function: \( \cos(\theta - 180^\circ) \). Using the cosine subtraction formula or the cosine shift identity, recognize that \( \cos(\alpha - 180^\circ) = -\cos(\alpha) \). So, \( \cos(\theta - 180^\circ) = -\cos(\theta) \).
Determine the sign of \( \cos(\theta) \) for \( \theta \) in the interval \( (-90^\circ, 90^\circ) \). Since cosine is positive in Quadrant I (0° to 90°) and positive in Quadrant IV (-90° to 0°), \( \cos(\theta) > 0 \) in this range.
Since \( \cos(\theta) > 0 \), then \( -\cos(\theta) < 0 \). Therefore, \( \cos(\theta - 180^\circ) \) is negative for \( \theta \) in the given interval.
Summarize: The sign of \( \cos(\theta - 180^\circ) \) is negative when \( -90^\circ < \theta < 90^\circ \).

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Angle Measurement and Quadrants

Angles in trigonometry are measured in degrees or radians and are positioned within four quadrants on the coordinate plane. Knowing that θ is between -90° and 90° places it in Quadrants I or IV, which helps determine the sign of trigonometric functions based on the angle's location.
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Quadratic Formula

Reference Angles and Angle Transformations

Transforming an angle by adding or subtracting 180° shifts it to the opposite side of the unit circle. For cos(θ - 180°), the angle moves to a quadrant opposite to θ, affecting the sign of the cosine value due to the symmetry and periodicity of trigonometric functions.
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Reference Angles on the Unit Circle

Sign of the Cosine Function in Different Quadrants

The cosine function is positive in Quadrants I and IV and negative in Quadrants II and III. Understanding which quadrant the transformed angle lies in allows us to determine whether cos(θ - 180°) is positive or negative based on the cosine sign rules.
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Sum and Difference of Sine & Cosine
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