摘要:极地坐标和参数等量

In this chapter you explored  polar forms of graphs and equations, as well as parameters and parametric equations.  
::在本章中,你们探讨了极形的图表和方程,以及参数和参数方程。

Polar graphs are useful, we discovered, for many applications in science—and you can just imagine their applications for mapping the globe.
::我们发现,极地图对科学的许多应用很有用, 你可以想象它们对于绘制地球图的应用。

Polar coordinates can be converted from rectangular coordinates, and equations can also be converted to and from polar form. Systems of polar equations can be solved just as systems in the rectangular plane. Conics can be converted for use in the polar plane as well. Complex numbers can be converted to polar form and graphed accordingly.  
::极坐标可以从矩形坐标转换为极坐标,方程式也可以转换为极形。极方程式的系统可以与矩形平面的系统一样解析。二次方程式也可以转换为极平面使用。复杂数字可以转换为极形,并据此绘制图表。

Operations with polar coordinates include the Product and Quotient theorems, as well as power and root applications, including De Moivre's Theorem. Applications of these theorems provide insight into the nature of divisors and roots of larger and larger numbers.
::具有极地坐标的操作包括产品和引号定理,以及动力和根应用程序,包括De Moivre的定理,这些定理的应用使人们深入了解了大数和大数的分解和根的性质。

Parametric equations involve multiple independent  variables within sets of equations that are dependent on each other. These equations  can help us model physical situations including movement, allowing us to add direction or orientation to our calculations.
::参数方程包含多个独立的变量,它们互相依赖。 这些方程可以帮助我们模拟物理环境,包括运动,让我们在计算中增加方向或方向。

Chapter Summary
::章次摘要

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• Points in the polar coordinate system are given in the form $\begin{array}{r}\left(r,\theta \right)\end{array}$ . The r -axis is referring to the radius r . To plot a specific point, first find the point that is r units from the origin on the r -axis. Then rotate counterclockwise by the given angle, commonly represented as " $\begin{array}{r}\theta \end{array}$ ."
  ::极地坐标系统中的点以窗体表示(r, ) 。 r- 轴指半径 r 。 要绘制一个特定的点, 请先从 r- 轴的源点找到 r 单位 。 然后以给定角度逆时旋转, 通常以“ \ ” 表示 。
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• We can use the formula  $\begin{array}{r}R=\frac{d\pi }{180},\end{array}$  where R is the number of radians and d is the number of degrees, to convert from radians to degrees or degrees to radians.
  ::我们可以使用公式R=d180, 其中R是弧度的数量, d是度的数量, 以便从弧度转换为度或度, 转换为弧度。
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• We can use graphing software, calculators, or plotting programs to plot polar equations.
  ::我们可以使用图形化软件、计算器或绘图程序来绘制极方程式。
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• We can convert coordinates from rectangular form to polar form.
  ::我们可以将方形坐标从矩形转换为极形。
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• We can also convert coordinates from polar form to rectangular form.
  ::我们还可以将坐标从极形转换为矩形。
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• We can solve a system of polar equations by plotting the equations on the same set of axes and determining their points of intersection.
  ::我们可以通过在同一组轴上绘制方程式并确定其交叉点,从而解决极地方程式系统的问题。
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• We can also solve a system of polar equations by setting the equations equal to each other and solving the resulting trigonometric equation.
  ::我们还可以通过设定等同方程式和解决由此产生的三角方程式来解决极地方程式系统。
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• Conic sections—including circles, ellipses, parabolas, and hyperbolas—have a common general polar equation.
  ::锥形块——包括圆圈、椭圆、parapolas和双螺旋——有一个共同的一般极等式。
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• In the standard form of  $\begin{array}{r}z=a+bi\end{array}$ , a complex number z can be graphed using rectangular coordinates ( a , b ), where ' a'  represents the x -coordinate, while ' b'  represents the y -coordinate.
  ::在z=a+bi的标准格式中,可使用矩形坐标(a,b)绘制复数z,其中“a”表示x坐标,“b”表示Y坐标。
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• U se x and y to convert between rectangular and polar forms with  $\begin{array}{r}r=\sqrt{x^2+y^2}\end{array}$ and $\begin{array}{r}\text{tan}{\theta }_{ref}=\left|\frac{y}{x}\right|\end{array}$ .
  ::使用 x 和 y 在 r= x2+y2 和 tan refyx 之间转换矩形和极形。
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• The  trigonometric form of  $\begin{array}{r}z=r(\cos \theta +i\sin \theta ),\end{array}$ which is often abbreviated as  rcisθ .
  ::三角形的z=r(cosisin), 通常缩写为 rcis。
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• The Product Theorem states  $\begin{array}{r}(r_{1}cis{\theta }_1)(r_{2}cis{\theta }_2)=r_1{r}_{2}cis({\theta }_1+{\theta }_2).\end{array}$
  
  ::产品定理表示(r1cis 1 (r2cis 2)=r1r2cis (1 2)。
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• The Quotient Theorem states that for $\begin{array}{r}{z}_1=r_1(cos{\theta }_1+isin{\theta }_1)\end{array}$  and $\begin{array}{r}{z}_1=r_1(cos{\theta }_1+isin{\theta }_1),\end{array}$   $\begin{array}{r}\frac{z_1}{z_2}=\frac{r_1}{r_2}cis({\theta }_1-{\theta }_2).\end{array}$
  
  ::引文理论称,对于z1=r1(cos1+isin1)和z1=r1(cos1+isin1),z1z2=r1r2cis(cos1+isin2),z1z2=r1r2cis(12)。
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• De Moivre's Theorem tells us that for  $\begin{array}{r}z=r(\cos \theta +i\sin \theta ),\end{array}$  a complex number in rcisθ  form, if n is a positive integer,  $\begin{array}{r}{z}^n=r^n(cos(n\theta )+isin(n\theta ))\end{array}$ .
  ::Deivre的Theorem告诉我们,对于z=r(cosisin),如果 n是正整数,在 rcis形式中是一个复杂的数字,zn=rn(cos+isin)。
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• Given that the formula for De Moivre's Theorem also works for fractional powers, the same formula can be used for finding roots:  $\begin{array}{r}{z}^{1/n}=(a+bi{)}^{1/n}=r^{1/n}cis\left(\frac{\theta }{n}\right).\end{array}$
  
  ::鉴于De Moivre的定理公式也适用于分数功率,在寻找根值时也可以使用相同的公式:z1/n=(a+bi)1/n=r1/ncis。
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• "Eliminating the parameter" is a phrase that means to turn a parametric equation that has $\begin{array}{r}x=f(t)\end{array}$  and $\begin{array}{r}y=g(t)\end{array}$  into just a relationship between $\begin{array}{r}y\end{array}$  and $\begin{array}{r}x\end{array}$ .
  ::“恢复参数”是指将具有 x=f(t) 和 y=g(t) 的参数方程转换为y和 x之间的关系。
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• Parametric form refers to a relationship that includes $\begin{array}{r}x=f(t)\end{array}$  and $\begin{array}{r}y=g(t)\end{array}$ .
  ::参数形式是指包含 x=f(t) 和 y=g(t) 的关系。
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• Parameterization also means writing or describing in parametric form.
  ::参数化还意味着以参数形式书写或描述。
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• Two functions are inverses if for every point $\begin{array}{r}(a,b)\end{array}$  on the 1st function, there exists a point $\begin{array}{r}(b,a)\end{array}$  on the 2nd function.
  ::如果对于第一个函数的每一个点(a,b),在第二个函数上存在一个点(b,a),则两个函数是反向的。
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• An intersection for two sets of parametric equations happens when the points exist at the same $\begin{array}{r}x\end{array}$ , $\begin{array}{r}y\end{array}$ , and $\begin{array}{r}t\end{array}$ .
  ::两组参数方程式的交叉点发生于点点位于相同的x、y和t。
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• Circular motion and projectile motion are two applications of parametric equations.
  ::圆形运动和投射运动是参数方程的两个应用。

Review
::回顾

Try the following cumulative review problems to practice the concepts we studied in this chapter:
::尝试下列累积审查问题,以实践我们在本章中研究的概念: