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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 102
Capitolo 2, Problema 102

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. sin[270° + n • 360°]

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Recognize that the expression inside the sine function is \(270° + n \cdot 360°\), where \(n\) is an integer. Since \$360°$ represents a full rotation, adding \(n \cdot 360°\) does not change the sine value due to the periodicity of the sine function.
Recall the periodicity property of sine: \(\sin(\theta + 360°) = \sin(\theta)\) for any angle \(\theta\). Therefore, \(\sin(270° + n \cdot 360°) = \sin(270°)\).
Evaluate \(\sin(270°)\). Using the unit circle, \$270°$ corresponds to the point $(0, -1)$, so \(\sin(270°) = -1\).
Conclude that for any integer \(n\), \(\sin(270° + n \cdot 360°) = -1\).
Therefore, the expression is equal to \(-1\) for all integer values of \(n\).

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Periodic Properties of the Sine Function

The sine function is periodic with a period of 360°, meaning sin(θ + 360°) = sin(θ) for any angle θ. This property allows simplification of expressions involving angles plus multiples of 360°, reducing them to equivalent angles within one full rotation.
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Period of Sine and Cosine Functions

Reference Angles and Quadrant Sign Determination

The value of sine depends on the angle's position in the unit circle. Angles like 270° correspond to specific points where sine takes values -1, 0, or 1. Understanding which quadrant or axis the angle lies on helps determine the sine value's sign and magnitude.
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Reference Angles on the Unit Circle

Evaluating Sine at Standard Angles

Certain angles such as 0°, 90°, 180°, 270°, and 360° have known sine values (0, 1, 0, -1, 0 respectively). Recognizing these standard angles allows direct evaluation of sine expressions without complex calculations.
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Drawing Angles in Standard Position
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