Course: 12.7 复杂数字极表

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 复数极表

Collapse Expand
复数极表
Introduction
::导言

can be graphed on a polar graph in a few different ways. This section discusses how to write a complex number in standard form, polar form , and trigonometric form . Then, based on these forms, it covers how to graph a complex number on the rectangular and polar coordinate systems.
::可以用几种不同的方式在极图中绘制图表。本节讨论如何以标准形式、极形式和三角形式写出一个复数。然后, 以这些形式为基础, 它涵盖了如何用矩形和极坐标系统绘制一个复数。

Polar Form of Complex Numbers
::复数极表

There are three common forms of complex numbers you will see when graphing:
::有三种常见的复杂数字形式, 图形化时您可以看到 :

1) The standard form of a complex number, $\begin{aligned} z = a + b i \end{aligned}$ , can be graphed using rectangular coordinates $\begin{aligned} (a, b), \end{aligned}$ where $\begin{aligned} a \end{aligned}$ represents the $\begin{aligned} x \end{aligned}$ -coordinate and $\begin{aligned} b \end{aligned}$ represents the $\begin{aligned} y \end{aligned}$ -coordinate.
::1) 复数标准格式z=a+bi,可以用矩形坐标(a,b)绘制图表,其中表示x坐标,b表示Y坐标。

2) The polar form, $\begin{aligned} (r, θ) \end{aligned}$ , can also be used to graph a complex number. Recall that you can convert between rectangular and polar forms with
$\begin{aligned} r = \sqrt{x^{2} + y^{2}} \end{aligned}$
and
$\begin{aligned} tan θ_{r e f} = | \frac{y}{x} | . \end{aligned}$
Unfortunately, there is a problem with using a conversion from rectangular form to polar form:
$\begin{aligned} a + b i \to (r, θ) \end{aligned}$
or
$\begin{aligned} - 1 - i \sqrt{3} \to (2, \frac{4 π}{3}) . \end{aligned}$
Notice that we no longer see the in this form.

3) The trigonometric form, which is often abbreviated as $\begin{aligned} r c i s θ \end{aligned}$ , comes from the substitutions $\begin{aligned} x = r \cos θ \end{aligned}$ and $\begin{aligned} y = r \sin θ \end{aligned}$ . Suppose $\begin{aligned} z = - 1 - i \sqrt{3} \end{aligned}$ . Note that in polar form, this complex number is $\begin{aligned} (2, \frac{4 π}{3}) \end{aligned}$ .

$\begin{aligned} z & = 2 \cos \frac{4 π}{3} + 2 i \sin \frac{4 π}{3} \\ = 2 (\cos \frac{4 π}{3} + i \sin \frac{4 π}{3}) \end{aligned}$

The complex number $\begin{aligned} z = - 1 - \sqrt{3} i \end{aligned}$ can be written as a rectangular point $\begin{aligned} (- 1, - \sqrt{3}) \end{aligned}$ , a polar point $\begin{aligned} (2, \frac{4 π}{3}) \end{aligned}$ , or in trigonometric form $\begin{aligned} 2 (cos \frac{4 π}{3} + i sin \frac{4 π}{3}) \end{aligned}$ or $\begin{aligned} 2 cis (\frac{4 π}{3}) \end{aligned}$ .
::2)极形,(r, ) 也可以用来绘制一个复杂的数字。提醒注意, 您可以用 r=x2+y2 和 tan ärefyx 来转换矩形和极形, r=x2+y2 和 tan ärefyx 。不幸地, 使用从矩形转换成极形有问题: a+bi(r, }) 或-1- i3-i3-i3 (cos443+is%43) 。 3 注意, 复数 z1-3i 可以写成矩形点(-1-3), 极点(2, 43) 和 y rsin 。假设z1-i3 。注意, 在极形中, z=2,43 + sin3 =2( 43) 或三角形 (43) 。

Conversion
::改划

To convert from r ectangular to polar form, the distance that the point (2, 2) is from the origin can be found by
::要从矩形向极形转换成极形,点(2,2)与源的距离可以由下列方式找到:

$\begin{aligned} r = \sqrt{x^{2} + y^{2}} \end{aligned}$
or
$\begin{aligned} r = \sqrt{2^{2} + 2^{2}} = \sqrt{8} = 2 \sqrt{2} . \end{aligned}$

::r= rx2+y2或r=22+22=8=22。

The reference angle (i.e. the corresponding angle in the 1st quadrant) that the line segment between the point and the origin can be found by using
::参考角(即第1象方图中相应的角),即点与源之间的线段通过使用

$\begin{aligned} tan θ_{r e f} = | \frac{y}{x} | . \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我...

For $\begin{aligned} z = 2 + 2 i, \end{aligned}$

$\begin{aligned} \tan θ_{ref} = \frac{2}{2} = 1. \end{aligned}$

::z=2+2i, tanref=22=1。

Since this point is in the 1st quadrant (both the x- and y- coordinates are positive), the angle must be 45° or $\begin{aligned} \frac{π}{4} \end{aligned}$ radians.
::由于此点位于第1象限( X 和 Y 坐标均为正),角必须是 45 ° 或 4 弧度。

It is also possible that when $\begin{aligned} \tan θ = 1, \end{aligned}$ the angle can be in the 3rd quadrant or $\begin{aligned} \frac{5 π}{4} \end{aligned}$ radians. However, this angle will not satisfy the conditions of the problem, since a 3rd-quadrant angle must have both $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ as negatives.
::当 tan1 1 时, 角也有可能在 3 象限或 5 4 弧度中。但是, 这个角无法满足问题的条件, 因为 3 象限角度必须同时有 x 和 y 的负值。

Note: When using $\begin{aligned} tan θ = \frac{y}{x} \end{aligned}$ , you should consider the quotient $\begin{aligned} | \frac{y}{x} | \end{aligned}$ and find the 1st-quadrant angle that satisfies this condition. This angle will be called the reference angle, denoted $\begin{aligned} θ_{r e f} \end{aligned}$ . Find the actual angle by analyzing which quadrant the angle must be in, given the signs of x and y .
::注意: 使用 tan yx 时, 您应该考虑 yx 的商数, 并找到符合此条件的 1- quamont 角度。这个角度将被称为参考角度 , 表示 ref 。如果 x 和 y 的标记, 则通过分析角必须进入的方位数来查找实际角度。

The complex number $\begin{aligned} 2 + 2 i \end{aligned}$ or (2, 2) in rectangular form has $\begin{aligned} (2 \sqrt{2}, \frac{π}{4}) \end{aligned}$ .
::2+2i或(2,2)的矩形复合体有(22,%4)。

The following video demonstrates how to write complex numbers in trigonometric form:
::以下视频展示如何以三角形式写出复杂的数字:

Examples
::实例

Example 1
::例1

Convert into polar form: $\begin{aligned} z = - 1 - i \sqrt{3} \end{aligned}$ .
::转换成极形: z1- i3。

Here is what it looks like in the rectangular coordinate system:
::以下是矩形坐标系统中它的样子:

Solution:
::解决方案 :

To convert $\begin{aligned} z = - 1 - i \sqrt{3} \end{aligned}$ to polar , first determine $\begin{aligned} r \end{aligned}$ and $\begin{aligned} θ \end{aligned}$ .
::将z1-i3转换为极地,首先确定 R和。

$\begin{aligned} r & = \sqrt{a^{2} + b^{2}} \\ = \sqrt{(- 1)^{2} + (- \sqrt{3})^{2}} \\ = \sqrt{1 + 3} \\ = \sqrt{4} \\ = 2 \end{aligned}$

::r=a2+b2=(-1)2+(-3)2=1+3=3=4=2

$\begin{aligned} \tan θ_{ref} & = | \frac{- \sqrt{3}}{- 1} | = \sqrt{3} \\ θ_{ref} & = \tan^{- 1} \sqrt{3} = \frac{π}{3} \end{aligned}$

::3133333333413333

Since this angle is in the 3rd quadrant, $\begin{aligned} θ = \frac{4 π}{3} \end{aligned}$ .
::由于这个角度位于第3象限, 43。

Example 2
::例2

Find the polar coordinates that represent the complex number $\begin{aligned} z = 3 - 3 \sqrt{3} i \end{aligned}$ .
::查找代表复数z=3-333i的极坐标。

Solution:
::解决方案 :

a = 3 and b = $\begin{aligned} - 3 \sqrt{3} \end{aligned}$ , so the rectangular coordinates of the point are $\begin{aligned} (3, - 3 \sqrt{3}) \end{aligned}$ .
::a = 3和b =-33,因此点的矩形坐标为(3,-33)。

Now, draw a right triangle in standard form. Find the distance the point is from the origin, and the angle the line segment that represents this distance makes with the +x axis:
::现在, 以标准格式绘制一个右三角形。查找点与源的距离, 以及代表此距离的线段与 +x 轴的角 :

We know a = 3, $\begin{aligned} b = - 3 \sqrt{3} \end{aligned}$ .
$\begin{aligned} r & = \sqrt{3^{2} + (- 3 \sqrt{3})^{2}} \\ = \sqrt{9 + 27} \\ = \sqrt{36} \\ = 6 \end{aligned}$

::我们知道a=3,b33.r=32+(-33)2=9+27=36=6

$\begin{aligned} \tan θ_{ref} & = | \frac{- 3 \sqrt{3}}{3} | = \sqrt{3} \\ θ_{ref} & = \tan^{- 1} \sqrt{3} = \frac{π}{3} \end{aligned}$

::$$333$3$3$3$3$3$3$3$3$3$3$3$3$3$3$

Since it is a 4th-quadrant angle, $\begin{aligned} θ = \frac{5 π}{3} \end{aligned}$ .
::因为它是第四赤道角 53

The rectangular point $\begin{aligned} (3, - 3 \sqrt{3} i) \end{aligned}$ is equivalent to the polar point $\begin{aligned} (6, \frac{5 π}{3}) \end{aligned}$ .
::矩形点(3,-33i)相当于极点(6,53)。

In trigonometric form, $\begin{aligned} (3, - 3 \sqrt{3} i) \end{aligned}$ is $\begin{aligned} 6 (cos \frac{5 π}{3} + i sin \frac{5 π}{3}) \end{aligned}$ .
::在三角形式中,(3,-33i)为6(5°3+i sin 53)。

Example 3
::例3

Plot the complex number $\begin{aligned} z = 12 + 9 i \end{aligned}$ .
::绘制复合号z=12+9i。

Solution:
::解决方案 :

To plot $\begin{aligned} z = 12 + 9 i \end{aligned}$ on a polar graph, first determine $\begin{aligned} r \end{aligned}$ and $\begin{aligned} θ \end{aligned}$ .
::要在极图上绘制 z=12+9i,首先确定 r 和。

$\begin{aligned} r & = \sqrt{12^{2} + 9^{2}} \\ = \sqrt{144 + 81} \\ = \sqrt{225} \\ = 15 \end{aligned}$

::r=122+92=144+81=225=15

$\begin{aligned} \tan θ_{ref} & = | \frac{9}{12} | \\ θ_{ref} & = \tan^{- 1} \frac{9}{12} \approx 36.9 ° \end{aligned}$

::912 -1 912 36.9°

$\begin{aligned} z = 12 + 9 i \end{aligned}$ looks like the image below when plotted on a polar plane.
::z=12+9i 绘制在极平面上时看起来像下面的图像。

Example 4
::例4

In what quadrant does $\begin{aligned} z = - 3 + 2 i \end{aligned}$ occur when graphed?
::当图表显示时, z3+2i 在什么象限中发生 ?

Solution:
::解决方案 :

The point $\begin{aligned} z = - 3 + 2 i \end{aligned}$ occurs 3 units to the left and 2 units up, placing it in quadrant 2.
::点 z3+2i 向左3个单位,向上2个单位,放在象限2中。

Example 5
::例5

What are the coordinates of $\begin{aligned} z = - 3 + 2 i \end{aligned}$ in polar form and trigonometric form?
::z3+2i 以极形和三角形表示的坐标是多少?

Solution:
::解决方案 :

To convert $\begin{aligned} z = - 3 + 2 i \end{aligned}$ , first determine $\begin{aligned} r \end{aligned}$ and $\begin{aligned} θ \end{aligned}$ .
::要转换 z3+2i,首先确定 R和。

$\begin{aligned} r = \sqrt{(- 3)^{2} + 2^{2}} = \sqrt{13} \end{aligned}$

::r=(-3)2+22=13

$\begin{aligned} \tan θ_{ref} & = | \frac{2}{- 3} | = \frac{2}{3} \\ θ_{ref} & = \tan^{- 1} \frac{2}{3} \approx {33.7}^{\circ} \end{aligned}$

::2323771237

Therefore, $\begin{aligned} (\sqrt{13}, {33.7}^{\circ}) \end{aligned}$ is the coordinate in polar form, and $\begin{aligned} \sqrt{13} c i s ({33.7}^{\circ}) \end{aligned}$ is the coordinate in trigonometric form.
::因此,(13,33.7)是极形的坐标,13cis(33.7)是三角形的坐标。

Example 6
::例6

What would be the polar coordinates of the point graphed below?
::下图显示的点的极地坐标是多少?

Solution:
::解决方案 :

The rectangular coordinates are (4.5, 3), so the complex number is $\begin{aligned} z = 4.5 + 3 i \end{aligned}$ .

$\begin{aligned} r = \sqrt{{4.5}^{2} + 3^{2}} \approx 5.4 \end{aligned}$

::矩形坐标是(4.5, 3), 所以复数是z=4.5+3i. r=4.52+325. 4。

$\begin{aligned} \tan θ_{ref} & = | \frac{3}{4.5} | \\ θ_{ref} & = \tan^{- 1} \frac{3}{4.5} \approx {33.7}^{\circ} \end{aligned}$

::34.555477

Therefore, $\begin{aligned} (5.4, {33.7}^{\circ}) \end{aligned}$ is the coordinate in polar form, and $\begin{aligned} 5.4 c i s ({33.7}^{\circ}) \end{aligned}$ is the coordinate in trigonometric form.
::因此,(54.33.7)是极形的坐标,5.4cis(33.7)是三角形的坐标。

Summary
::摘要

In the standard form of $\begin{aligned} z = a + b i, \end{aligned}$ a complex number z can be graphed using rectangular coordinates $\begin{aligned} (a, b) \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ represents the $\begin{aligned} x \end{aligned}$ -coordinate, while $\begin{aligned} b \end{aligned}$ represents the $\begin{aligned} y \end{aligned}$ -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的图表,其中代表 x 坐标,而b 代表 Y 坐标。

Use x and y to convert between rectangular and polar forms with $\begin{aligned} r = \sqrt{x^{2} + y^{2}} \end{aligned}$ and $\begin{aligned} tan θ_{r e f} = | \frac{y}{x} | \end{aligned}$ .
::使用 x 和 y 在 r= x2+y2 和 tan refyx 之间转换矩形和极形。

Review
::回顾

Plot each complex number below in the complex plane. Convert its polar form $\begin{aligned} (r, θ), \end{aligned}$ where $\begin{aligned} θ \end{aligned}$ is in degrees.
::在复杂的平面下绘制每个复数。转换它的极形( r, ) , 其中 ° 以度表示。

$\begin{aligned} 1 + i \end{aligned}$
::1个+一

$\begin{aligned} (1 + i) i \end{aligned}$
:1+一)i

$\begin{aligned} (- 2) (3 i) \end{aligned}$
:-2)(3)(一)

$\begin{aligned} 1 + i \end{aligned}$
::1个+一

$\begin{aligned} 1 - i \end{aligned}$
::1-一

$\begin{aligned} (1 + i) (1 - i) \end{aligned}$
:1+一)(1)-(一)

$\begin{aligned} 1 + i \sqrt{3} \end{aligned}$
::1+i3 1+i3

$\begin{aligned} \sqrt{3} - i \end{aligned}$
::3- i 3- i

$\begin{aligned} (1 + i \sqrt{3}) (\sqrt{3} - i) \end{aligned}$
:1+i3)(3-i)

What are the rectangular coordinates for the point graphed below?
::下图点的矩形坐标是多少?

C onvert to rectangular form:
::转换为矩形形式 :

$\begin{aligned} 15 (c o s 120^{o} + i s i n 120^{o}) \end{aligned}$
::15(cos120o+isin120o)

$\begin{aligned} 12 (c o s \frac{π}{3} + i s i n \frac{π}{3}) \end{aligned}$
::12(cos%3+isin%3) 3

For the complex number in standard form $\begin{aligned} x + i y, \end{aligned}$ find: a) polar form, and b) trigonometric form. (Hint: Recall that $\begin{aligned} x = r c o s θ \end{aligned}$ and $\begin{aligned} y = r s i n θ . \end{aligned}$ )
::对于标准表x+iy的复数,请找到:a)极形,和b)三角形。 (提示:提醒注意 x=rcos- 和 y=rsin- 。 )

Find the zeros of $\begin{aligned} f (x) = x^{2} - 4 x + 8. \end{aligned}$ Graph them on the complex plane.
::查找 f( x) =x2-4x+8. 的零。在复合平面上绘制图。

Convert to trigonometric form:
::转换为三角形 :

$\begin{aligned} 1 + i^{6} \end{aligned}$
::1+i6 1+i6

$\begin{aligned} \frac{\sqrt{3}}{2} + \frac{1}{2} i^{10} \end{aligned}$
::32+12i10

$\begin{aligned} - 3 - 2 i \end{aligned}$
::- 3-2i

$\begin{aligned} 2 \sqrt{3} - 2 i \end{aligned}$
::23-2i 23-2ii

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Resource: Graphing Calculator Instructions
::资源: 图形计算器指令

Convert the following complex numbers into polar form, us ing technology:

a) $\begin{aligned} \sqrt{3} - i \end{aligned}$

b) $\begin{aligned} 9 \sqrt{3} + 9 i \end{aligned}$

::使用技术将下列复杂数字转换成极形a) 3-i(b) 93+9i

Solution
::解决方案

On the TI-84, go to [ANGLE] (or [2nd] function) [APPS] . Scroll down to 5 or "R-Pr(" and press [Enter] . Next, enter the rectangular coordinates and close the parentheses. Press [Enter]; the "r" value appears. Scroll down to 6R-Pθ, and the polar angle appears in decimal radian form.
::在 TI-84 上, 转到 [ANGLE] (或 [第 2 函数) [APPS] 。向下滚动到 5 或“ R- Pr () ” , 按 [Enter] 。下一步, 输入矩形坐标并关闭括号。按 [Enter] ; 显示“ r” 值。向下滚动到 6R- P , 极角以小数弧形显示。

Note: Also under the [ANGLE] menu, commands 7 and 8 allow transformation from polar form to rectangular form.
::注:在[ANGLE]菜单下,指令7和8允许从极形转换成矩形。

Section outline

General

复数极表

Introduction ::导言

Polar Form of Complex Numbers ::复数极表

Examples ::实例

Summary ::摘要

Review ::回顾

Review (Answers) ::回顾(答复)

Resource: Graphing Calculator Instructions ::资源: 图形计算器指令

Convert the following complex numbers into polar form, us ing technology: a) 3 − i b) 9 3 + 9 i ::使用技术将下列复杂数字转换成极形a) 3-i(b) 93+9i

Solution ::解决方案

Introduction
::导言

Polar Form of Complex Numbers
::复数极表

Examples
::实例

Summary
::摘要

Review
::回顾

Review (Answers)
::回顾(答复)

Resource: Graphing Calculator Instructions
::资源: 图形计算器指令

Convert the following complex numbers into polar form, us ing technology:

a) $\begin{aligned} \sqrt{3} - i \end{aligned}$

b) $\begin{aligned} 9 \sqrt{3} + 9 i \end{aligned}$

::使用技术将下列复杂数字转换成极形a) 3-i(b) 93+9i

Solution
::解决方案