sinusoidal function

简明释义

正弦函数

英英释义

例句

1.The temperature variations throughout the day can often be modeled with a sinusoidal function (正弦函数).

一天中的温度变化通常可以用一个sinusoidal function (正弦函数) 来建模。

2.Sound waves can be described as sinusoidal functions (正弦函数) in physics.

在物理学中，声波可以被描述为sinusoidal functions (正弦函数)。

3.In electrical engineering, alternating current is represented by a sinusoidal function (正弦函数).

在电气工程中，交流电由sinusoidal function (正弦函数) 表示。

4.The graph of a sinusoidal function (正弦函数) shows a smooth, periodic oscillation.

一个sinusoidal function (正弦函数) 的图形显示出平滑的周期性振荡。

5.The motion of a pendulum can be described using a sinusoidal function (正弦函数).

摆的运动可以用一个sinusoidal function (正弦函数) 来描述。

作文

In the realm of mathematics, particularly in trigonometry and calculus, the concept of a sinusoidal function is fundamental. A sinusoidal function is a mathematical function that describes a smooth, periodic oscillation. The most common examples of sinusoidal functions are the sine and cosine functions, which are defined using the unit circle. These functions are not only crucial in pure mathematics but also have significant applications in various fields such as physics, engineering, and signal processing.To understand a sinusoidal function, one must first grasp its graphical representation. When plotted on a coordinate system, a sinusoidal function produces a wave-like pattern that repeats at regular intervals, known as the period. The height of the wave at any point is referred to as the amplitude, which indicates the maximum value of the function from its central axis. For instance, the sine function oscillates between -1 and 1, creating a wave that peaks at these values. This characteristic behavior makes sinusoidal functions ideal for modeling periodic phenomena such as sound waves, light waves, and even seasonal changes in temperature.The mathematical expression for a sinusoidal function can be written in the form of y = A sin(B(x - C)) + D, where A represents the amplitude, B affects the period, C is the phase shift, and D is the vertical shift. Each of these parameters plays a crucial role in shaping the graph of the function. For example, increasing the amplitude A will stretch the wave vertically, while altering B will change the frequency of the oscillation. Understanding these parameters allows mathematicians and scientists to manipulate sinusoidal functions to fit real-world data accurately.Beyond their mathematical beauty, sinusoidal functions serve practical purposes. In physics, they describe oscillatory motions, such as the swinging of a pendulum or the vibrations of a guitar string. Engineers use sinusoidal functions in the design of circuits and systems that involve alternating current (AC), where the voltage varies sinusoidally over time. Furthermore, in signal processing, sinusoidal functions are used to analyze and synthesize signals, enabling technologies such as audio compression and telecommunications.In addition to these applications, the study of sinusoidal functions provides insights into harmonic analysis, which examines how complex waveforms can be decomposed into simpler sine and cosine waves. This principle is foundational in Fourier analysis, a mathematical tool widely used in engineering, physics, and applied mathematics.In conclusion, the sinusoidal function is a powerful concept that extends beyond the confines of mathematics. Its periodic nature and ability to model real-world phenomena make it invaluable in various scientific and engineering disciplines. Whether one is analyzing sound waves, designing electrical systems, or exploring the intricacies of harmonic motion, understanding sinusoidal functions is essential. As we continue to delve deeper into the world of mathematics and its applications, the importance of sinusoidal functions remains ever-present, reminding us of the beauty and utility of mathematical concepts in our everyday lives.

在数学领域，特别是三角学和微积分中，正弦函数的概念是基础。正弦函数是描述平滑、周期性振荡的数学函数。最常见的正弦函数示例是正弦和余弦函数，它们使用单位圆定义。这些函数不仅在纯数学中至关重要，而且在物理学、工程学和信号处理等多个领域也有重要应用。要理解正弦函数，首先必须掌握其图形表示。当在坐标系上绘制时，正弦函数会产生一种波浪状的图案，在规则的间隔内重复，称为周期。波浪在任何点的高度称为幅度，它表示函数从其中心轴的最大值。例如，正弦函数在-1和1之间振荡，形成一条在这些值处达到峰值的波。这种特征行为使得正弦函数非常适合建模周期现象，例如声波、光波，甚至温度的季节变化。正弦函数的数学表达式可以写成y = A sin(B(x - C)) + D的形式，其中A代表幅度，B影响周期，C是相位移，D是垂直移位。每个参数在塑造函数图形方面都起着至关重要的作用。例如，增加幅度A将垂直拉伸波，而改变B将改变振荡的频率。理解这些参数使得数学家和科学家能够准确地操纵正弦函数以适应现实世界的数据。除了数学之美外，正弦函数还具有实际用途。在物理学中，它们描述了振荡运动，例如摆的摇摆或吉他弦的振动。工程师在设计涉及交流电（AC）的电路和系统时使用正弦函数，其中电压随时间呈正弦变化。此外，在信号处理领域，正弦函数用于分析和合成信号，使音频压缩和电信等技术成为可能。除了这些应用外，研究正弦函数提供了对谐波分析的深入了解，后者研究复杂波形如何分解为更简单的正弦和余弦波。这一原理是傅里叶分析的基础，傅里叶分析是一种广泛用于工程、物理和应用数学的数学工具。总之，正弦函数是一个强大的概念，超越了数学的界限。它的周期性特征和模拟现实世界现象的能力使其在各种科学和工程学科中不可或缺。无论是分析声波、设计电气系统，还是探索谐波运动的复杂性，理解正弦函数都是必不可少的。随着我们继续深入研究数学及其应用，正弦函数的重要性始终存在，提醒我们数学概念在日常生活中的美丽和实用性。