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. cos[n • 360°]

검증된 단계별 안내

Recognize that the expression is \( \cos[n \cdot 360^\circ] \), where \( n \) is an integer.

Recall the periodicity of the cosine function: \( \cos(\theta) = \cos(\theta + 360^\circ) \) for any angle \( \theta \).

Since \( n \) is an integer, \( n \cdot 360^\circ \) represents full rotations around the unit circle, landing back at the starting point.

Therefore, \( \cos[n \cdot 360^\circ] = \cos(0^\circ) \), because rotating by full circles does not change the cosine value.

Recall that \( \cos(0^\circ) = 1 \), so the expression evaluates to 1 for any integer \( n \).

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Periodic Nature of the Cosine Function

The cosine function is periodic with a period of 360°, meaning cos(θ) = cos(θ + 360°k) for any integer k. This property allows simplification of angles by reducing them modulo 360°, which is essential for evaluating expressions like cos(n • 360°).

Integer Multiples of Angles

When an angle is expressed as an integer multiple of a base angle (e.g., n • 360°), it helps identify patterns in trigonometric values. For cosine, multiples of 360° correspond to full rotations, which return the function to its initial value, simplifying evaluation.

Values of Cosine at Special Angles

Cosine values at key angles such as 0°, 90°, 180°, 270°, and 360° are well-known: cos(0°) = 1, cos(90°) = 0, cos(180°) = -1, etc. Recognizing these values helps determine the output of cosine expressions involving multiples of these angles, like cos(n • 360°) = 1.