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[n • 180°]

Key Concepts

Sine Function and Periodicity

The sine function, denoted as sin(x), is a periodic function with a period of 360°. This means that sin(x) = sin(x + 360°k) for any integer k. Specifically, sin(n • 180°) will yield values based on the periodicity of the sine function, where sin(180°) = 0 and sin(0°) = 0, leading to sin(n • 180°) being 0 for any integer n.

Integer Multiples of Angles

When dealing with integer multiples of angles, such as n • 180°, we can simplify the sine function by recognizing that these angles correspond to specific points on the unit circle. For any integer n, n • 180° will always land on the x-axis of the unit circle, where the sine value is 0, reinforcing that sin(n • 180°) = 0.

Trigonometric Values at Key Angles

Trigonometric functions have specific values at key angles, such as 0°, 90°, 180°, and 270°. For the sine function, sin(0°) = 0, sin(90°) = 1, and sin(180°) = 0. Since n • 180° results in angles that are multiples of 180°, the sine of these angles will consistently yield 0, which is crucial for evaluating the expression sin[n • 180°].