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°]

검증된 단계별 안내

Recall the general behavior of the sine function: \(\sin(\theta)\) equals 0 whenever \(\theta\) is an integer multiple of \(180^\circ\), i.e., \(\theta = k \times 180^\circ\) for any integer \(k\).

Given the expression \(\sin[n \cdot 180^\circ]\), recognize that \(n\) is an integer, so the angle is exactly an integer multiple of \(180^\circ\).

Use the property of sine at these angles: \(\sin(k \times 180^\circ) = 0\) for all integers \(k\).

Therefore, \(\sin[n \cdot 180^\circ]\) must be equal to 0 for any integer \(n\).

No undefined values or other outputs (like 1 or -1) occur for this expression because sine is defined for all real numbers and specifically zero at these multiples.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sine Function and Its Periodicity

The sine function is periodic with a period of 360°, meaning sin(θ) = sin(θ + 360°k) for any integer k. This periodicity helps simplify angles by reducing them modulo 360°, making it easier to evaluate sine values for large or multiple-angle expressions.

Sine of Integer Multiples of 180°

For any integer n, sin(n • 180°) equals zero because 180° corresponds to π radians, where the sine function crosses the x-axis. Thus, sin(n • 180°) = 0 for all integers n, reflecting the zeros of the sine wave at these points.

Angle Representation in Degrees and Radians

Angles can be expressed in degrees or radians, with 180° equal to π radians. Understanding this conversion is essential for interpreting trigonometric expressions and applying known sine values at key angles, such as multiples of 90° or 180°.