unit circle

简明释义

单位圆

英英释义

例句

1.To find the coordinates of a point on the unit circle 单位圆, you can use the cosine and sine of the angle.

要找到单位圆 unit circle上某个点的坐标，可以使用该角度的余弦和正弦值。

2.When studying periodic functions, the unit circle 单位圆 is an essential tool.

在研究周期函数时，单位圆 unit circle是一个重要工具。

3.In trigonometry, the angles are often measured in radians using the unit circle 单位圆.

在三角学中，角度通常使用单位圆 unit circle以弧度来测量。

4.The unit circle 单位圆 helps to visualize how the sine and cosine functions relate to angles.

通过单位圆 unit circle可以帮助我们可视化正弦和余弦函数与角度的关系。

5.The unit circle 单位圆 is defined as a circle with a radius of one centered at the origin.

单位圆 unit circle被定义为一个半径为1且中心在原点的圆。

作文

The concept of the unit circle is fundamental in trigonometry and plays a crucial role in understanding the relationships between angles and the coordinates of points on a plane. A unit circle is defined as a circle with a radius of one, centered at the origin of a coordinate system. This simple yet powerful tool allows us to visualize and compute the values of trigonometric functions such as sine, cosine, and tangent for various angles. To illustrate the importance of the unit circle, consider how we can use it to find the sine and cosine of an angle. When an angle is drawn in standard position, its initial side lies along the positive x-axis, and its terminal side is determined by the angle's measure. The intersection of the terminal side with the unit circle gives us a point (x, y), where x represents the cosine of the angle and y represents the sine of the angle. Thus, every angle corresponds to a unique point on the unit circle, making it easier to understand these trigonometric functions. Moreover, the unit circle allows for the determination of trigonometric values for angles beyond the first quadrant. By extending our understanding to all four quadrants, we can see how the signs of sine and cosine change based on the location of the angle. For example, in the second quadrant, the sine value is positive while the cosine value is negative. This property is essential when working with angles greater than 90 degrees or less than 0 degrees. Another significant aspect of the unit circle is its application in periodic functions. Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The unit circle helps us visualize this periodicity and understand how angles that differ by full rotations (360 degrees or 2π radians) yield the same sine and cosine values. For instance, both 30 degrees and 390 degrees correspond to the same point on the unit circle, which allows us to conclude that sin(30°) = sin(390°) and cos(30°) = cos(390°). Furthermore, the unit circle serves as a bridge between algebra and geometry. It provides a geometric interpretation of trigonometric identities and equations, making it easier for students to grasp these concepts. For example, the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1, can be visually understood by recognizing that any point (x, y) on the unit circle satisfies this equation, since x² + y² = 1 for a circle of radius one. In conclusion, the unit circle is an invaluable tool in mathematics that enhances our understanding of trigonometric functions, their properties, and their applications. By providing a visual representation of angles and their corresponding sine and cosine values, the unit circle simplifies complex concepts and fosters a deeper comprehension of trigonometry. Whether one is solving equations, graphing functions, or exploring the relationships between angles, the unit circle remains an essential component of mathematical education and practice.

单位圆的概念在三角学中是基础性的，并且在理解角度与平面上点的坐标之间的关系中起着至关重要的作用。单位圆被定义为一个半径为一、以坐标系的原点为中心的圆。这种简单而强大的工具使我们能够可视化和计算各种角度的三角函数值，例如正弦、余弦和正切。为了说明单位圆的重要性，考虑如何使用它来找到一个角度的正弦和余弦。当一个角以标准位置绘制时，其初始边沿着正x轴，而其终端边由角度的度量确定。终端边与单位圆的交点给我们一个点(x, y)，其中x代表角度的余弦，y代表角度的正弦。因此，每个角度对应于单位圆上的一个唯一点，这使得理解这些三角函数变得更加容易。此外，单位圆允许我们确定位于第一象限之外的角度的三角函数值。通过将我们的理解扩展到所有四个象限，我们可以看到正弦和余弦的符号如何根据角度的位置变化。例如，在第二象限，正弦值为正，而余弦值为负。这一特性在处理大于90度或小于0度的角度时至关重要。单位圆的另一个重要方面是其在周期函数中的应用。三角函数是周期性的，这意味着它们在规律的间隔内重复其值。单位圆帮助我们可视化这种周期性，并理解不同完整旋转（360度或2π弧度）的角度如何产生相同的正弦和余弦值。例如，30度和390度都对应于单位圆上的同一点，这使我们得出结论：sin(30°) = sin(390°) 和 cos(30°) = cos(390°)。此外，单位圆作为代数与几何之间的桥梁。它提供了三角恒等式和方程的几何解释，使学生更容易掌握这些概念。例如，毕达哥拉斯恒等式，即sin²(θ) + cos²(θ) = 1，可以通过认识到任何在单位圆上的点(x, y)都满足此方程，因为对于半径为一的圆，有x² + y² = 1，来进行视觉理解。总之，单位圆是数学中一项无价的工具，它增强了我们对三角函数、它们的性质及其应用的理解。通过提供角度及其对应的正弦和余弦值的可视化表示，单位圆简化了复杂的概念，并促进了对三角学的更深入理解。无论是在解方程、绘制函数图形，还是探索角度之间的关系，单位圆始终是数学教育和实践的重要组成部分。