### 计算机代写|计算机图形学作业代写computer graphics代考|Trigonometry

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

The measurement of angles is at the heart of trigonometry, and today two units of angular measurement are part of modern mathematics: degrees and radians. The degree (or sexagesimal) unit of measure derives from defining one complete rotation as $360^{\circ}$. Each degree divides into $60 \mathrm{~min}$, and each minute divides into $60 \mathrm{~s}$. The number 60 has survived from Mesopotamian days and appears rather incongruous when used alongside today’s decimal system – nevertheless, it is still convenient to work with degrees even though the radian is a natural feature of mathematics.

The radian of angular measure does not depend upon any arbitrary constant, and is often defined as the angle created by a circular arc whose length is equal to the circle’s radius. And because the perimeter of a circle is $2 \pi r, 2 \pi$ rad correspond to one complete rotation. As $360^{\circ}$ corresponds to $2 \pi \mathrm{rad}, 1 \mathrm{rad}$ equals $180^{\circ} / \pi$, which is approximately $57.3^{\circ}$. The following relationships between radians and degrees are

worth remembering:
$\frac{\pi}{2}[\mathrm{rad}] \equiv 90^{\circ}, \quad \pi[\mathrm{rad}] \equiv 180^{\circ}$
$\frac{3 \pi}{2}[\mathrm{rad}] \equiv 270^{\circ}, \quad 2 \pi[\mathrm{rad}] \equiv 360^{\circ} .$
To convert $x^{\circ}$ to radians:
$$\frac{\pi x^{\circ}}{180}[\mathrm{rad}]$$
To convert $x[\mathrm{rad}]$ to degrees:
$$\frac{180 x}{\pi} \text { [degrees]. }$$
For those readers wishing to know the background to radians we need to use power series. We start with the power series for $\mathrm{e}^{\theta}, \sin \theta$ and $\cos \theta$ :
\begin{aligned} \mathrm{e}^{\theta} &=1+\frac{\theta^{1}}{1 !}+\frac{\theta^{2}}{2 !}+\frac{\theta^{3}}{3 !}+\frac{\theta^{4}}{4 !}+\frac{\theta^{5}}{5 !}+\frac{\theta^{6}}{6 !}+\frac{\theta^{7}}{7 !}+\frac{\theta^{8}}{8 !}+\frac{\theta^{9}}{9 !}+\cdots \ \sin \theta &=\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !}-\frac{\theta^{7}}{7 !}+\frac{\theta^{9}}{9 !}+\cdots \ \cos \theta &=1-\frac{\theta^{2}}{2 !}+\frac{\theta^{4}}{4 !}-\frac{\theta^{6}}{6 !}+\frac{\theta^{8}}{8 !}+\cdots \end{aligned}
Euler proved that these three power series are related, and when $\theta=\pi, \sin \theta=0$, and $\cos \theta=-1$. Figure $4.1$ shows curves of the sine power series for $3,5,7$ and 9 terms, and when $\theta=2 \pi$, the graph reaches zero.

## 计算机代写|计算机图形学作业代写computer graphics代考|The Trigonometric Ratios

Ancient civilisations knew that triangles-whatever their size-possessed some inherent properties, especially the ratios of sides and their associated angles. This means that if these ratios are known in advance, problems involving triangles with unknown lengths and angles, can be discovered using these ratios.

Figure $4.2$ shows a point $P$ with coordinates (base, height), on a unit-radius circle rotated through an angle $\theta$. As $P$ is rotated, it moves into the 2 nd quadrant, 3rd quadrant, 4th quadrant and returns back to the first quadrant. During the rotation, the sign of height and base change as follows:
$$\begin{array}{ll} \text { 1st quadrant: } & \text { height }(+) \text {, base }(+) \ \text { 2nd quadrant: } & \text { height }(+) \text {, base }(-) \ \text { 3rd quadrant: } & \text { height }(-) \text {, base }(-) \ \text { 4th quadrant: } & \text { height }(-) \text {, base }(+) . \end{array}$$
Figures $4.3$ and $4.4$ plot the changing values of height and base over the four quadrants, respectively. When radius $=1$, the curves vary between 1 and $-1$. In the context of triangles, the sides are labelled as follows:
\begin{aligned} \text { hypotenuse } &=\text { radius } \ \text { opposite } &=\text { height } \ \text { adjacent } &=\text { base. } \end{aligned}
Thus, using the right-angle triangle shown in Fig. 4.5, the trigonometric ratios: sine, cosine and tangent are defined as
$$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}, \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}, \quad \tan \theta=\frac{\text { opposite }}{\text { adjacent }} .$$

## 计算机代写|计算机图形学作业代写computer graphics代考|Inverse Trigonometric Ratios

The functions $\sin \theta, \cos \theta, \tan \theta, \csc \theta, \sec \theta$ and $\cot \theta$ provide different ratios for the angle $\theta$, and the inverse trigonometric functions convert a ratio back into an angle. These are arcsin, arccos, arctan, arccsc, arcsec and arccot, and are sometimes written as $\sin ^{-1}, \cos ^{-1}, \tan ^{-1}, \csc ^{-1}, \sec ^{-1}$ and $\cot ^{-1}$. For example, $\sin 30^{\circ}=0.5$, therefore, $\arcsin 0.5=30^{\circ}$. Consequently, the domain for arcsin is the range for sin:
$[-1,1]$
and the range for arcsin is the domain for sin:
$$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$
as shown in Fig. 4.8. Similarly, the domain for arccos is the range for cos:
$$[-1,1]$$
and the range for arccos is the domain for $\cos$ :
$$[0, \pi]$$
as shown in Fig. 4.9.

## 计算机代写|计算机图形学作业代写computer graphics代考|Units of Angular Measurement

3圆周率2[r一种d]≡270∘,2圆周率[r一种d]≡360∘.

180X圆周率 [度]。

## 计算机代写|计算机图形学作业代写computer graphics代考|The Trigonometric Ratios

第一象限：  高度 (+)， 根据 (+)  第二象限：  高度 (+)， 根据 (−)  第三象限：  高度 (−)， 根据 (−)  第四象限：  高度 (−)， 根据 (+).

斜边 = 半径   对面的 = 高度   邻近的 = 根据。

## 计算机代写|计算机图形学作业代写computer graphics代考|Inverse Trigonometric Ratios

[−1,1]
arcsin 的范围是 sin 的域：
[−圆周率2,圆周率2]

[−1,1]
arccos 的范围是因 :
[0,圆周率]

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## 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。