addition theorem

简明释义

1. 加法定理; 2. 可加性定理;

英英释义

例句

1.The professor explained the addition theorem 加法定理 during the lecture on probability theory.

教授在概率论的讲座中解释了addition theorem 加法定理。

2.In statistics, the addition theorem 加法定理 assists in calculating the probabilities of combined events.

在统计学中，addition theorem 加法定理 有助于计算组合事件的概率。

3.The addition theorem 加法定理 is crucial for understanding the behavior of sine and cosine functions.

理解正弦和余弦函数的行为时，addition theorem 加法定理 是至关重要的。

4.In advanced mathematics, the addition theorem 加法定理 helps in deriving the sum of angles in trigonometric functions.

在高级数学中，addition theorem 加法定理 有助于推导三角函数的角度和。

5.Using the addition theorem 加法定理, we can simplify complex algebraic expressions.

利用addition theorem 加法定理，我们可以简化复杂的代数表达式。

作文

The concept of the addition theorem is fundamental in various fields of mathematics, particularly in geometry and trigonometry. To understand its significance, we must first explore what the addition theorem entails. In simple terms, the addition theorem refers to a principle that relates the sum of angles to the sine and cosine functions. This theorem allows us to express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles. For instance, if we have two angles A and B, the addition theorem states that: \[ \sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B \] \[ \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \] These formulas are not only elegant but also incredibly useful in solving problems involving triangles and waves. In fact, the addition theorem is often employed in physics, engineering, and computer graphics to analyze wave behavior and to perform transformations.To illustrate the practical application of the addition theorem, consider a scenario where an engineer needs to calculate the resultant vector when two forces are applied at an angle to each other. By using the addition theorem, the engineer can break down the forces into their respective components and then apply the sine and cosine functions to find the magnitude and direction of the resultant force. This method not only simplifies the calculations but also enhances accuracy, which is crucial in engineering design.Moreover, the addition theorem extends beyond traditional mathematics into areas such as statistics and probability. In these fields, the theorem helps in understanding the behavior of random variables and in deriving important distributions. For example, in the context of independent random variables, the addition theorem can be used to find the expected value of the sum of these variables, which is essential for making predictions based on data.In conclusion, the addition theorem is more than just a mathematical formula; it is a powerful tool that facilitates problem-solving across various disciplines. Whether it’s in calculating angles in a triangle, analyzing forces in physics, or predicting outcomes in statistics, the addition theorem proves to be invaluable. As students and professionals alike continue to delve into the world of mathematics and its applications, a solid understanding of the addition theorem will undoubtedly enhance their analytical skills and contribute to their success in their respective fields.

“加法定理”的概念在数学的各个领域中都是基础，特别是在几何和三角学中。要理解其重要性，我们首先必须探讨一下“加法定理”的内容。简单来说，“加法定理”是指一个原理，它将角度的和与正弦和余弦函数联系起来。这个定理使我们能够将两个角的和的正弦和余弦表示为单个角的正弦和余弦。例如，如果我们有两个角A和B，“加法定理”指出：\[ \sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B \] \[ \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \] 这些公式不仅优雅，而且在解决涉及三角形和波动的问题时非常有用。实际上，“加法定理”常常被应用于物理、工程和计算机图形学中，以分析波动行为和进行变换。为了说明“加法定理”的实际应用，考虑一个工程师需要计算当两个力以一定角度施加时的结果向量的场景。通过使用“加法定理”，工程师可以将力分解为各自的分量，然后应用正弦和余弦函数来找到结果力的大小和方向。这种方法不仅简化了计算，而且提高了准确性，这在工程设计中至关重要。此外，“加法定理”超越了传统数学，扩展到了统计学和概率论等领域。在这些领域中，该定理有助于理解随机变量的行为，并推导出重要的分布。例如，在独立随机变量的背景下，“加法定理”可以用来找到这些变量和的期望值，这对于基于数据进行预测至关重要。总之，“加法定理”不仅仅是一个数学公式；它是一个强大的工具，促进了各个学科的问题解决。无论是在计算三角形中的角度、分析物理中的力，还是在统计中预测结果，“加法定理”都被证明是无价的。随着学生和专业人士继续深入探索数学及其应用，扎实理解“加法定理”无疑会增强他们的分析能力，并为他们在各自领域的成功做出贡献。