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# 平面几何代写plane geometry代考|MAT116 Law of Cosines and Law of Sines

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## 数学代写|复几何代写Complex Geometry代考|Law of Cosines and Law of Sines

The Law of Cosines is the relationship between the sides and the angles of a triangle: $a^2=b^2+c^2-2 b c \cdot \cos (\angle A)$, where $\angle A$ is opposite of side $a$. Students usually learn this formula at the end of a geometry course but don’t understand how to apply it until taking trigonometry. Nevertheless, this formula is very powerful. In addition, the Pythagorean Theorem is a particular case of the Law of Cosines. Let us now solve Problem 8 :

Problem 8. A triangle with sides of length 5, 12, and 13 is given. Find the angle that is opposite the biggest side.

Solution. First, we will draw an accurate picture and put all known information on it so it will help us to calculate the measure of angle $C$ (Fig. 1.15).

Let us apply the Law of Cosines to the triangle:
\begin{aligned} &13^2=5^2+12^2-2 \cdot 5 \cdot 12 \cdot \cos (\angle C) \ &0=\cos (\angle C), \quad \text { then } \quad m \angle C=90^{\circ} \end{aligned}
We found that triangle $A B C$ is a right triangle.
Let us prove that the Pythagorean Theorem is a particular case of the Law of Cosines. For this purpose we will write the Law of Cosines for the cosine of angle $C$ using the same picture:
\begin{aligned} &c^2=a^2+b^2-2 a b \cos (\angle C) \ &\cos (\angle C)=\frac{a^2+b^2-c^2}{2 a b} \end{aligned}

## 数学代写|复几何代写Complex Geometry代考|Similar Triangles

Two triangles are called similar if their corresponding angles are equal and the ratio of corresponding sides is the same. Similar triangles or, in general, any similar figures have a similar shape. You probably remember that two triangles are similar to each other by two angles $(A A)$, by two sides and the included angle $(S A S)$, and by three sides (SSS). Especially important is the fact that in similar triangles the ratio of corresponding sides, medians, heights, and bisectors equals $k$, the coefficient of similitude. The ratio of the areas of similar triangles equals $k^2$, the square of the coefficient of similitude.

The following picture will illustrate one way of constructing similar triangles by hand, with $k>1$ (magnification) and $k<1$ (making the image smaller). This is based on Homothetic Transformation or such transformation at which any point of the original (pre-image), its image and the center of homothety lie on the same line. Using homothety, different similar figures can be constructed, so all corresponding angles are equal.

## 数学代写|复几何代写Complex Geometry代考|Law of Cosines and Law of Sines

$$13^2=5^2+12^2-2 \cdot 5 \cdot 12 \cdot \cos (\angle C) \quad 0=\cos (\angle C) \text {, then } m \angle C=90^{\circ}$$

$$c^2=a^2+b^2-2 a b \cos (\angle C) \quad \cos (\angle C)=\frac{a^2+b^2-c^2}{2 a b}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。