### 数学代写|微积分代写Calculus代写|Trigonometry Review

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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

## 数学代写|微积分代写Calculus代写|Angles

One way of defining an angle is to say that it is the union of two rays, as pictured in figure 1. It is common to consider angles in standard position, with the vertex at the origin and the initial side along the positive $x$-axis (figure 2). The other ray is called the terminal side of the angle. Initial and terminal conjure the idea of movement, and we can think of the ray rotating around the vertex from the initial side to the terminal side. If the rotation is counterclockwise, then the angle has a positive measure; if the rotation is clockwise, then the angle has a negative measure. Angles are sometimes labeled near the vertex between the initial and terminal sides.

The two common units of measure used to describe the size of an angle are degrees and radians. Although most students of trigonometry find degrees easier to use than radians, many calculations in calculus

are much easier if radians are used instead of degrees for measuring angles.

One radian is the size of an angle that subtends an arc of the same length as the radius of the circle. For instance, in a circle of radius 1 , an angle of 1 radian (rad) captures an arc of length 1 , as in figure 3 . The sector pictured in figure 3 looks almost like an equilateral triangle, but with one side curved. Angles in an equilateral triangle measure $60^{\circ}$, but adding a slight curve to the side opposite the angle reduces the size of the angle to a little under $60^{\circ}$ (to approximately $57.3^{\circ}$ ).

Double the angle, and the arc it subtends is doubled (figure 4 , left). Halve the angle, and the arc it subtends is halved (figure 4 , middle). In a circle of radius 1 , an angle of $\theta \mathrm{rad}$ subtends an arc of length $\theta$ (figure 4 , right).Still using a circle of radius 1 , what happens if we make one full revolution? The angle measures $360^{\circ}$. In radians, the measure of the angle equals the measure of the arc all the way around the circle, which is the circle’s circumference. The circumference of a circle is $2 \pi r$, so a circle of radius 1 has circumference $2 \pi \cdot 1=2 \pi$. Therefore, the angle measures $2 \pi$ rad. See figure 5 .

Example 1 Convert the measurement $135^{\circ}$ to radians.
Solution Using the conversion factor $1^{\circ}=\frac{\pi}{180} \mathrm{rad}$, we have
\begin{aligned} 135 \cdot 1^{\circ} &=135 \cdot \frac{\pi}{180} \mathrm{rad} \ 135^{\circ} &=\frac{135}{180} \pi \mathrm{rad}=\frac{3 \pi}{4} \mathrm{rad} . \end{aligned}
The measurement $135^{\circ}$ is equal to $\frac{3 \pi}{4} \mathrm{rad}$.
It should be obvious that not having instant recall of arithmetic facts such as $2+4=6$ is a huge impediment to learning algebra. Imagine simplifying $2 x+4 x$ while having to count on one’s fingers to reach the result $6 x$, counting on fingers every time arithmetic is needed. Working algebra problems would be interminably slow! The same is true of the need for instant recall of certain facts in trigonometry. Not having instant recall of facts such as $30^{\circ}=\frac{\pi}{6} \mathrm{rad}$ or $\frac{3 \pi}{2} \mathrm{rad}=270^{\circ}$ can easily turn a 1-hour calculus assignment into a 3-hour assignment. It can also make the difference between finishing an exam with ample time to review work and not having time to attempt all the problems, leaving some blank.

Among the facts needing instant recall are the following comparisons between degrees and radians, and the resulting angle in standard position (table 1).

## 数学代写|微积分代写Calculus代写|Trigonometric functions and their values

Among the methods of defining the trigonometric functions is the following. Begin with an angle $\theta$ in standard position, measured in radians. Choose a point $(x, y)$ on the terminal side of the angle and drop a perpendicular from the point $(x, y)$ to the $x$-axis, forming a right triangle. Label the horizontal leg of the triangle $x$, the vertical leg $y$, and the hypotenuse $r$. These steps are illustrated in figure 11 , from left to right.

Notice that by labeling the legs of the triangle $x$ and $y$, the numbers are allowed to be negative. In fact, if the terminal side of the triangle lies along a coordinate axis, the value of $x$ or $y$ can also be zero. Because we have a right triangle, by the Pythagorean theorem,
$$r=\sqrt{x^{2}+y^{2}} .$$
Notice that $r$ is always positive and never negative. Using the diagram on the far right side of figure 11 , we are now ready to define the six trigonometric functions, which represent the six possible ratios of sides of the triangle.

The names of the trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant, with emphasis always on the first syllable. The abbreviations are pronounced by their function names. For instance, “sin $\theta$ ” is pronounced “sine theta” and “csc $\theta$ ” is pronounced “cosecant theta.”

As an aid to memorizing the six trig ratios, notice that each row in definition 5 contains one “co-” function (top row, cosecant; second row, cosine; third row, cotangent). The two ratios in each row are reciprocals. Whether the ratios are memorized by their letters $(x, y, r)$ or by their descriptions (adjacent, opposite, hypotenuse; see figure 12) is a matter of personal preference.

Example $3 A$ point on the terminal side of the angle $\theta$ (in standard position) is $(5,-12)$. Find the values of the six trigonometric functions.
Solution We use $x=5, y=-12$, and
\begin{aligned} r &=\sqrt{x^{2}+y^{2}} \ &=\sqrt{5^{2}+(-12)^{2}} \ &=\sqrt{25+144} \ &=\sqrt{169}=13 \end{aligned}
It is helpful to draw the triangle and label its sides, so that a visual connection is made between the trig functions and their formulas and values (figure 13).

## 数学代写|微积分代写Calculus代写|Angles

1 弧度是对着与圆的半径相同长度的弧的角度的大小。例如，在半径为 1 的圆中，1 弧度 (rad) 的角度捕获长度为 1 的弧，如图 3 所示。图 3 中描绘的扇区看起来几乎像一个等边三角形，但一侧是弯曲的。等边三角形测量中的角度60∘, 但是在角度的对面添加一个轻微的曲线可以将角度的大小减小到一点点以下60∘（到大约57.3∘ ).

135⋅1∘=135⋅圆周率180r一个d 135∘=135180圆周率r一个d=3圆周率4r一个d.

## 数学代写|微积分代写Calculus代写|Trigonometric functions and their values

r=X2+是2.

r=X2+是2 =52+(−12)2 =25+144 =169=13

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## MATLAB代写

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