trigonometric function

简明释义

三角函数

英英释义

例句

1.To solve this triangle, we can use the sine and cosine trigonometric functions to find the missing angles and sides, which are 三角函数.

要解决这个三角形，我们可以使用正弦和余弦三角函数来找到缺失的角和边，属于三角函数.

2.The graph of a trigonometric function like sine or cosine exhibits periodic behavior, which is a characteristic of 三角函数.

像正弦或余弦这样的三角函数的图形表现出周期性行为，这是三角函数的一个特征.

3.In calculus, we often encounter the derivative of a trigonometric function, which is known as a 三角函数.

在微积分中，我们经常会遇到一个三角函数的导数，它被称为trigonometric function.

4.When calculating the height of a building, architects often rely on trigonometric functions to determine angles and distances, which involve 三角函数.

在计算建筑物的高度时，建筑师通常依赖于三角函数来确定角度和距离，涉及到三角函数.

5.In physics, trigonometric functions are used to describe oscillations and waves, which can be represented by 三角函数.

在物理学中，三角函数用于描述振荡和波动，这可以用三角函数表示.

作文

Trigonometric functions are essential components of mathematics, especially in the field of geometry and calculus. These functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions include sine, cosine, and tangent, which are foundational for understanding various mathematical concepts. In this essay, I will explore the significance of trigonometric functions in both theoretical and practical applications, demonstrating their importance in our everyday lives.To begin with, let us define what trigonometric functions are. They are functions that arise from the ratios of the sides of a right triangle. For instance, in a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Similarly, cosine is the ratio of the adjacent side to the hypotenuse, while tangent is the ratio of the opposite side to the adjacent side. These definitions not only help in solving triangles but also serve as a foundation for more advanced mathematical studies.One of the most remarkable aspects of trigonometric functions is their periodic nature. The sine and cosine functions, for example, repeat their values in a regular pattern, which can be observed in their graphs. This periodicity makes them incredibly useful in modeling phenomena that exhibit cyclical behavior, such as sound waves, light waves, and even seasonal changes in temperature. By using trigonometric functions, scientists and engineers can predict and analyze these patterns effectively.Moreover, trigonometric functions play a critical role in calculus, particularly in integration and differentiation. For instance, the derivatives of sine and cosine functions are fundamental results that are frequently used in various applications. Understanding how to manipulate these functions allows mathematicians to solve complex problems involving rates of change and areas under curves. In physics, trigonometric functions are employed to describe oscillatory motion, such as the swinging of a pendulum or the vibration of a string.In addition to their theoretical applications, trigonometric functions have numerous practical uses in fields such as engineering, architecture, and computer graphics. Engineers use these functions to design structures and analyze forces acting on them. Architects rely on trigonometric functions to create blueprints that ensure buildings are stable and aesthetically pleasing. In computer graphics, trigonometric functions help create realistic animations and simulations by allowing for the representation of circular and oscillatory movements.Furthermore, the advent of technology has made it easier to visualize and understand trigonometric functions. Graphing calculators and software programs allow students and professionals to plot these functions, providing a clear representation of their behavior. This visual aid enhances comprehension and facilitates learning, especially for those who may struggle with abstract mathematical concepts.In conclusion, trigonometric functions are indispensable tools in mathematics, science, and engineering. Their ability to relate angles to side lengths in triangles, along with their periodic properties, enables us to model and analyze a wide range of real-world phenomena. As we continue to advance in technology and mathematics, the significance of trigonometric functions will undoubtedly remain prominent, shaping our understanding of the world around us and enhancing our problem-solving capabilities.

三角函数是数学中不可或缺的组成部分，尤其是在几何和微积分领域。这些函数将三角形的角度与其边的长度联系起来。主要的三角函数包括正弦、余弦和正切，这些是理解各种数学概念的基础。在这篇文章中，我将探讨三角函数在理论和实际应用中的重要性，展示它们在我们日常生活中的重要性。首先，让我们定义一下什么是三角函数。它们是从直角三角形的边的比率中产生的函数。例如，在一个直角三角形中，某个角的正弦被定义为对边长度与斜边长度的比率。类似地，余弦是邻边与斜边的比率，而正切是对边与邻边的比率。这些定义不仅有助于解决三角形问题，而且为更高级的数学研究奠定了基础。三角函数最显著的特点之一是它们的周期性。正弦和余弦函数，例如，以规律的模式重复它们的值，这可以在它们的图形中观察到。这种周期性使它们在建模表现出周期性行为的现象时极其有用，例如声波、光波甚至季节温度变化。通过使用三角函数，科学家和工程师可以有效地预测和分析这些模式。此外，三角函数在微积分中也扮演着关键角色，特别是在积分和微分方面。例如，正弦和余弦函数的导数是经常用于各种应用的基本结果。理解如何操作这些函数使数学家能够解决涉及变化率和曲线下面积的复杂问题。在物理学中，三角函数被用来描述振荡运动，例如摆的摆动或弦的振动。除了理论应用外，三角函数在工程、建筑和计算机图形等领域也有无数实际用途。工程师使用这些函数来设计结构并分析作用在它们上的力。建筑师依赖于三角函数来创建确保建筑物稳定且美观的蓝图。在计算机图形中，三角函数通过允许表示圆形和振荡运动来帮助创建逼真的动画和模拟。此外，科技的进步使可视化和理解三角函数变得更加容易。图形计算器和软件程序使学生和专业人员能够绘制这些函数，提供清晰的行为表示。这种可视化辅助增强了理解，促进了学习，尤其是对于那些可能在抽象数学概念上挣扎的人。总之，三角函数是数学、科学和工程中不可或缺的工具。它们将角度与三角形的边长联系起来的能力，加上它们的周期特性，使我们能够建模和分析广泛的现实世界现象。随着我们在科技和数学上的不断进步，三角函数的重要性无疑将继续突出，塑造我们对周围世界的理解，并增强我们的解决问题的能力。