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°]

Verified step by step guidance

1. 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.
2. Recall the periodicity property of sine: \(\sin(\theta + 360°) = \sin(\theta)\) for any angle \(\theta\). Therefore, \(\sin(270° + n \cdot 360°) = \sin(270°)\).
3. Evaluate \(\sin(270°)\). Using the unit circle, \$270°\( corresponds to the point \)(0, -1)$, so \(\sin(270°) = -1\).
4. Conclude that for any integer \(n\), \(\sin(270° + n \cdot 360°) = -1\).
5. Therefore, the expression is equal to \(-1\) for all integer values of \(n\).

Key Concepts

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.

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.

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.