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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

Given the value of one trig function of an angle and the quadrant in which the angle lies, the values of the other trig functions can be determined as well.

Example 9 Given $\sin \theta=\frac{4}{7}$ and $\frac{\pi}{2}<\theta<\pi$, find the other five trig functions of $\theta$.

Solution There are always two quadrants in which a given trig function has positive values. Knowing that $\sin \theta$ is a positive number is not enough to know in which quadrant the terminal side of $\theta$ lies.

We are told that the angle $\theta$ is a second-quadrant angle $(\theta$ is between $\frac{\pi}{2}$ and $\pi$ ). We begin by drawing an angle with the terminal side in the second quadrant, then we choose a point on the terminal side and drop a perpendicular to form a right triangle (figure 21 , left).Since $\sin \theta=\frac{y}{r}=\frac{4}{7}$, we label the vertical leg of the triangle 4 and the hypotenuse 7 (figure 21, right). The label for the horizontal leg, $x$, should be negative because we are in the second quadrant.

## 数学代写|微积分代写Calculus代写|Basic trig graphs

Points on the unit circle have coordinates $(\cos \theta, \sin \theta)$, as in figure 25 (right). Points on the graphs of $y=\sin \theta$ and $y=\cos \theta$ have the form $(\theta, \sin \theta)$ and $(\theta, \cos \theta)$, respectively (figure 25 , left). Tracing the points on the unit circle and the graphs as we allow $\theta$ to vary helps us see the relationship between the unit circle, values of $\sin \theta$ and $\cos \theta$, and their graphs. Start with $\theta=0$. The coordinates on the unit circle are $(1,0)$; therefore, $\cos 0=1$ and $\sin 0=0$. As $\theta$ gets larger, the point on the unit circle moves left and upward, meaning that the $x$-coordinate $(\cos \theta)$ decreases and the $y$-coordinate $(\sin \theta)$ increases. Eventually, as $\theta$ reaches $\frac{\pi}{2}$, the point on the unit circle reaches $(0,1)$, meaning $\cos \frac{\pi}{2}=0$ and $\sin \frac{\pi}{2}=1$. This is illustrated in figure 26 .

Continuing toward $\theta=\pi$ moves the point on the unit circle left and downward; therefore, $\cos \theta$ is becoming more negative and $\sin \theta$ is decreasing, eventually reaching $(-1,0)$, meaning that $\cos \pi=-1$ and $\sin \pi=0 .$ See figure 27 As we move toward $\theta=\frac{3 \pi}{2}$ and the point $(0,-1)$ on the unit circle, the $x$-coordinate moves right toward 0 and the $y$-coordinate moves more negative toward $-1$. Then, as $\theta$ moves toward $2 \pi$ and the point $(1,0)$ on the unit circle, the $x$-coordinate continues moving to the right toward 1 and the $y$-coordinate starts moving upward toward 0 . Then the entire process repeats, over and over, ad infinitum, every length of $2 \pi$. We may move clockwise into negative values of $\theta$, with the pattern still repeating. The final results are in figure 28 .

# 微积分代考

## 有限元方法代写

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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。

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