课程： 11.3 复杂数字的三角极表

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复杂数字的三角极表

You already know how to represent in the complex plane using rectangular coordinates and you already know how to multiply and divide complex numbers. Representing these points and performing these operations using trigonometric polar form will make your computations more efficient.
::您已经知道如何使用矩形坐标在复杂的平面上显示, 您已经知道如何乘和除以复杂的数字。代表这些点并使用三角极表进行这些操作将会提高计算的效率。

What are the two ways to multiply the following complex numbers?
::乘以以下复杂数字的两种方法是什么?

$\begin{aligned} (1 + \sqrt{3} i) (\sqrt{2} - \sqrt{2} i) \end{aligned}$
:1+3i)(2-2-2i)

Trigonometric Polar Form of Complex Numbers
::复杂数字的三角极表

Any point represented in the complex plane as $\begin{aligned} a + b i \end{aligned}$ can be represented in polar form just like any point in the rectangular coordinate system. Trigonometric polar form of a complex number describes the location of a point on the complex plane using the angle and the radius of the point. You will use the distance from the point to the origin as $\begin{aligned} r \end{aligned}$ and the angle that the point makes as $\begin{aligned} θ \end{aligned}$ .
::以 a+b 表示的复杂平面中的任何点可以像矩形坐标系统中的任何点一样以极形表示。复数的三角极形表示复杂平面上一个点的位置,使用角度和点半径表示。您将点与点之间的距离用作起点 r 和点作为角的角。

As you can see, the point $\begin{aligned} a + b i \end{aligned}$ can also be represented as $\begin{aligned} r \cdot \cos θ + i \cdot r \cdot \sin θ \end{aligned}$ . The trigonometric polar form can be abbreviated by factoring out the $\begin{aligned} r \end{aligned}$ and noting the first letters:
::您可以看到, a+bi 点也可以作为 rácosirsin 表示。三角极形可以通过分计 r 和注意前几个字母来缩略 :

$\begin{aligned} r (\cos θ + i \cdot \sin θ) \to r \cdot cis θ \end{aligned}$
:cosisin) r_cisis

The abbreviation $\begin{aligned} r \cdot cis θ \end{aligned}$ is read as “ $\begin{aligned} r \end{aligned}$ kiss theta .” It allows you to represent a point as a radius and an angle.
::缩略语 r-cis 被解读为“r kiss theta ” 。它允许您以半径和角度代表点。

Take the following complex number in rectangular form .
::以矩形形式取下列复数。

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

To convert the following complex number from rectangular form to trigonometric polar form, find the radius using the absolute value of the number.
::要将以下复数从矩形形转换为三角极形,请使用数字的绝对值查找半径。

$\begin{aligned} r^{2} = 1^{2} + {(- \sqrt{3})}^{2} \to r = 2 \end{aligned}$
::r2=12+(-3)2r=2

The angle can be found with basic trig and the knowledge that the opposite side is always the imaginary component and the adjacent side is always the real component.
::可以用基本三角法找到角度,并且知道对立方始终是假想的成分,而相邻方始终是真实的成分。

$\begin{aligned} \tan θ = - \frac{\sqrt{3}}{1} \to θ = 60^{\circ} \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我爱她 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我爱她 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我爱她 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

Thus the trigonometric form is $\begin{aligned} 2 cis 60^{\circ} \end{aligned}$ .
::因此三角形是2cis 60。

One great benefit of th e $\begin{aligned} c i s \end{aligned}$ form is that it makes multiplying and dividing complex numbers extremely easy. For example:
::精子形式的一大好处是,它使复杂数字的乘法和除法非常容易。例如:

Let: $\begin{aligned} z_{1} = r_{1} \cdot cis θ_{1}, z_{2} = r_{2} \cdot cis θ_{2} \end{aligned}$ with $\begin{aligned} r_{2} \neq 0 \end{aligned}$ .
::让我们:z1=r1=r1=r2=r2=r2=r2=r2=r2=r2=r2=r1z2=r2=r2=r2=r2=r2=r2=r1=r1=r1z2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r2=r=r2=r2=r2=r2=r2=r2=r2=r=r=r2=r2=r=r=r=r=r=r2=r2=r=r=r=r2=r=r=r=r=r2=r2=r=r=r2=r2=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=2=r=r=r=r=r=r=2=2=2=2=2=2=2=2=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=r=2=2=r=r=r=r=r=r=r=r=

Then:
::然后:

\begin{aligned} z_{1} \cdot z_{2} & = r_{1} \cdot r_{2} \cdot cis (θ_{1} + θ_{2}) \\ z_{1} \div z_{2} & = \frac{r_{1}}{r_{2}} \cdot cis (θ_{1} - θ_{2}) \end{aligned}

::z2=r1r2-cis(12)z1 @z2=r1r2_cis(12)

For basic problems, the amount of work required to compute products and quotients for complex numbers given in either form is roughly equivalent. For more challenging questions, trigonometric polar form becomes significantly advantageous.
::对于基本问题来说,计算产品和计算两种形式的复杂数字的商数所需的工作量大致相等,对于更具挑战性的问题,三角极形变得非常有利。

Examples
::实例

Example 1
::例1

Earlier, you were asked how to multiply the complex numbers $\begin{aligned} (1 + \sqrt{3} i) (\sqrt{2} - \sqrt{2} i) \end{aligned}$ .
::早些时候,有人问您如何乘以复数(1+3i)(2-2i)。

In rectangular coordinates:
::矩形坐标:

$\begin{aligned} (1 + \sqrt{3} i) (\sqrt{2} - \sqrt{2} i) = \sqrt{2} - \sqrt{2} i + \sqrt{6} i + \sqrt{6} \end{aligned}$
:1+3i)(2-2-2i)=2-2i+6i+6)

In trigonometric , $\begin{aligned} 1 + \sqrt{3} i = 2 cis 60^{\circ} \end{aligned}$ and $\begin{aligned} \sqrt{2} - \sqrt{2} i = 2 cis - 45^{\circ} \end{aligned}$ . Therefore:
::在三角测量中,1+3i=2cis 60和2-2i=2cis-45。

$\begin{aligned} (1 + \sqrt{3} i) (\sqrt{2} - \sqrt{2} i) = 2 cis 60^{\circ} \cdot 2 cis - 45^{\circ} = 4 cis 105^{\circ} \end{aligned}$
:1+3i)(2-2i)=2 Cis 602 Cis-45}4 Cis 105

Example 2
::例2

Convert the following complex number from trigonometric polar form to rectangular form.
::将以下的复数从三角极形转换成矩形形。

$\begin{aligned} 4 cis (\frac{3 π}{4}) \end{aligned}$
::4 Cis( 3__ 4)

$\begin{aligned} 4 cis (\frac{3 π}{4}) = 4 (\cos (\frac{3 π}{4}) + i \cdot \sin (\frac{3 π}{4})) = 4 (- \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i) = - 2 \sqrt{2} + 2 \sqrt{2} i \end{aligned}$
::4Cis (34) = 4(cos(34) +isin (34) = 4(- 22+22i) 22+22i)

Example 3
::例3

Divide the following complex numbers.
::除以下列复数。

$\begin{aligned} \frac{4 cis 32^{\circ}}{2 cis 2^{\circ}} \end{aligned}$
::4cis 322cis 2cis 2

$\begin{aligned} \frac{4 cis 32^{\circ}}{2 cis 2^{\circ}} = \frac{4}{2} cis (32^{\circ} - 2^{\circ}) = 2 cis (30^{\circ}) \end{aligned}$
::4cis 322cis 242cis(322)=2cis(30)

Example 4
::例4

Translate the following complex number from rectangular form into trigonometric polar form:
::将下列复数从矩形形转换成三角极形:

$\begin{aligned} 8 = 8 cis 0^{\circ} \end{aligned}$
::8=8 cis 0

Note that this has no complex part and therefore has no angle.
::请注意,这不是一个复杂的部分,因此没有角度。

Example 5
::例5

Multiply the following complex numbers in trigonometric polar form.
::以三角极形式乘以下列复数。

$\begin{aligned} 4 cis 34^{\circ} \cdot 5 cis 16^{\circ} \cdot \frac{1}{2} cis 100^{\circ} \end{aligned}$
::4C34Q5CS 16C12CS 100CS 4CS 34QQ5CS 16CS 12CS 100CS

\begin{aligned} 4 cis 34^{\circ} \cdot 5 cis 16^{\circ} \cdot \frac{1}{2} cis 100^{\circ} \\ = 4 \cdot 5 \cdot \frac{1}{2} \cdot cis (34^{\circ} + 16^{\circ} + 100^{\circ}) = 10 cis 150 \end{aligned}

::4 cis 345 cis 1612 cis 1004512cis (3416100) =10 cis 150

Note how much easier it is to do products and quotients in trigonometric polar form.
::注意以三角极形式制作产品和商数要容易得多。

Summary

Trigonometric polar form of a complex number describes the location of a point on the complex plane using the angle and the radius of the point.
::复数的三角极形式说明使用该点的角度和半径在复杂平面上点的位置。

The point $\begin{aligned} a + b i \end{aligned}$ can be represented as $\begin{aligned} r (\cos θ + i \cdot s i n θ) \end{aligned}$ which is abbreviated as $\begin{aligned} r \cdot cis θ \end{aligned}$
::点a+bi可以作为r(cosisin)表示,缩写为r__cis。

To convert a complex number from rectangular form to trigonometric polar form, find the radius using the absolute value of the number and the angle using basic trigonometry.
::要将复数从矩形形转换为三角极形,请使用数字的绝对值和基本三角法的角度来找到半径。

Review
::回顾

Translate the following complex numbers from trigonometric polar form to rectangular form.
::将以下的复杂数字从三角极形转换为矩形。

1. $\begin{aligned} 5 cis 270^{\circ} \end{aligned}$
::1. 5 CIS 270________________________________________________________________________________________________________________

2. $\begin{aligned} 2 cis 30^{\circ} \end{aligned}$
::2. 2个CIS 30_______________________________________________________________________________________________________________________________________________________________________

3. $\begin{aligned} - 4 cis \frac{π}{4} \end{aligned}$
::3. - 4 Cis 4

4. $\begin{aligned} 6 cis \frac{π}{3} \end{aligned}$
::4. 6 CIS 3

5. $\begin{aligned} 2 cis \frac{5 π}{2} \end{aligned}$
::5. 2 CIS 5°2

Translate the following complex numbers from rectangular form into trigonometric polar form.
::将以下复数从矩形形转换成三角极形。

6. $\begin{aligned} 2 - i \end{aligned}$
::2-一

7. $\begin{aligned} 5 + 12 i \end{aligned}$
::7. 5+12i

8. $\begin{aligned} 6 i + 8 \end{aligned}$
::8. 6i+8

9. $\begin{aligned} i \end{aligned}$
::9. i

Complete the following calculations and simplify.
::完成以下计算并简化。

10. $\begin{aligned} 2 cis 22^{\circ} \cdot \frac{1}{5} cis 15^{\circ} \cdot 3 cis 95^{\circ} \end{aligned}$
::10. 2个Cis 22 15个Cis 15 3个Cis 95个cis 95个cis

11. $\begin{aligned} 9 cis 98^{\circ} \div 3 cis 12^{\circ} \end{aligned}$
::11. 9个Cis 98 3个Cis 12个cis 12个cis

12. $\begin{aligned} 15 cis \frac{π}{4} \cdot 2 cis \frac{π}{6} \end{aligned}$
::12. 15支Cis 42支Cis 6

13. $\begin{aligned} - 2 cis \frac{2 π}{3} \div 15 cis \frac{7 π}{6} \end{aligned}$
::13. -2 CIS 2315 CIS 76

Let $\begin{aligned} z_{1} = r_{1} \cdot cis θ_{1} \end{aligned}$ and $\begin{aligned} z_{2} = r_{2} \cdot cis θ_{2} \end{aligned}$ with $\begin{aligned} r_{2} \neq 0 \end{aligned}$ .
::让 z1 = r1Cis%1 和 z2=r2Cis%2 加上 r2°0 。

14. Use the trigonometric sum and difference identities to prove that $\begin{aligned} z_{1} \cdot z_{2} = r_{1} \cdot r_{2} \cdot cis (θ_{1} + θ_{2}) \end{aligned}$ .
::14. 使用三角数和差异身份来证明z1z2=r1r2&cis(12)。

15. Use the trigonometric sum and difference identities to prove that $\begin{aligned} z_{1} \div z_{2} = \frac{r_{1}}{r_{2}} \cdot cis (θ_{1} - θ_{2}) \end{aligned}$ .
::15. 使用三角数和差异身份来证明z1z2=r1r2cis(12)。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

复杂数字的三角极表

Trigonometric Polar Form of Complex Numbers ::复杂数字的三角极表

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)