### 数学代写|微积分代写Calculus代写|The values of the six trig functions

statistics-lab™ 为您的留学生涯保驾护航 在代写微积分Calculus方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微积分Calculus代写方面经验极为丰富，各种代写微积分Calculus相关的作业也就用不着说。

• 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

## 数学代写|微积分代写Calculus代写|The values of the six trig functions

The values of the six trig functions are
$$\begin{array}{ll} \sin \theta=\frac{-12}{13} & \csc \theta=\frac{13}{-12} \ \cos \theta=\frac{5}{13} & \sec \theta=\frac{13}{5} \ \tan \theta=\frac{-12}{5} & \cot \theta=\frac{5}{-12} . \end{array}$$
Developing the habit of writing the trig function values in three rows and two columns as presented in example 3 is helpful. When the ratios in the first column are applied, the values in the second column are written quickly as the reciprocals of those in the first column.
No matter which point on the terminal side of the angle is used (figure 14), the values of the trig functions are the same. This is important, for we would not want two different values for $\sin \theta$, as that would make it fail to be a function. Because the results are the same no matter which point is chosen, we say that the trig functions are well-defined.

Two “special triangles” from geometry are helpful for quickly determining the values of trig functions: the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle and the $30^{\circ}-60^{\circ}-90^{\circ}$ triangle. With angles labeled in radians, these triangles are pictured in figure 15 . To use these triangles effectively, they must be on instant recall.

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).
Figure 21 Steps in drawing the diagram for example 9
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.

The value of $x$ may be found using the Pythagorean theorem:
\begin{aligned} x^{2}+4^{2} &=7^{2} \ x^{2}+16 &=49 \ x^{2} &=33 \ x &=-\sqrt{33} . \end{aligned}
We finish the diagram by labeling the horizontal leg (figure 22 ).
Figure 22 The completed figure for example 9
We are now ready to determine the values of the other five trig functions using the definition:
$$\begin{array}{ll} \cos \theta=\frac{-\sqrt{33}}{7} & \csc \theta=\frac{7}{4} \ \tan \theta=\frac{4}{-\sqrt{33}} & \sec \theta=\frac{7}{-\sqrt{33}} \ \cot \theta=\frac{-\sqrt{33}}{4} . \end{array}$$

## 数学代写|微积分代写Calculus代写|Useful trig identities

There are a very large number of trigonometric identities, equations that relate trig functions to one another and are true for all values for which the functions are defined. Some identities are much more commonly used than others, and the ability to recognize the applicability of an identity readily can be crucial to completing some calculus exercises. The identities described next are the ones that all calculus students should know.

The first set of identities includes the reciprocal identities. Since $\sin \theta=\frac{y}{r}$ and $\csc \theta=\frac{r}{y}$, the two trig functions are reciprocals of one another.

Notice that
$$\frac{\sin \theta}{\cos \theta}=\frac{\frac{y}{x}}{\frac{x}{r}}=\frac{y}{x}=\tan \theta .$$
This is called a ratio identity.
RATIO IDENTITIES
For any angle $\theta$ for which the functions are defined,
$$\tan \theta=\frac{\sin \theta}{\cos \theta} \quad \cot \theta=\frac{\cos \theta}{\sin \theta}$$
The reciprocal identities for secant and cosecant along with the ratio identities for tangent and cotangent give us the ability to write every trigonometric function in terms of sines and cosines. This ability can be very helpful in certain calculus exercises.

Example 10 Rewrite the expression $\tan \theta \csc \theta+2 \sec \theta$ in terms of sines and cosines.
Solution Using the reciprocal and ratio identities,
\begin{aligned} \tan \theta \csc \theta+2 \sec \theta &=\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}+2 \cdot \frac{1}{\cos \theta} \ &=\frac{1}{\cos \theta}+\frac{2}{\cos \theta} \ &=\frac{3}{\cos \theta} . \end{aligned}

## 数学代写|微积分代写Calculus代写|The values of the six trig functions

X2+42=72 X2+16=49 X2=33 X=−33.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。