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Chapter 3: Problem 10

Bestimmen Sie s?mtliche Lösungen: a) \(\quad z^{3}=64\left[\cos \left(\frac{\pi}{4}\right)+j \cdot \sin \left(\frac{\pi}{4}\right)\right]\) b) \(z^{3}-2=5 \mathrm{j}\)

Short Answer

Expert verified
a) Solutions are approximately \(z_0 = 3.85 + j1.03\), \(z_1 = -2.83 + j2.83\), \(z_2 = -1.03 - j3.85\). b) Solutions involve solving for \(\arctan(\frac{5}{2})\) and finding corresponding roots.

Step by step solution

01

- Express the equation in polar form (a)

Rewrite the given equation as \[ z^3 = 64 \left(\cos \frac{\pi}{4} + j \sin \frac{\pi}{4} \right) \]
02

- Identify magnitude and argument (a)

Identify the magnitude and argument of the complex number:\[ r^3 = 64 \quad \text{and} \quad 3\theta = \frac{\pi}{4} + 2k\pi \quad (k \in \mathbb{Z}) \]
03

- Solve for magnitude (a)

Solve for the magnitude:\[ r = \sqrt[3]{64} = 4 \]
04

- Solve for the argument (a)

Solve for the argument: \[ \theta = \frac{\frac{\pi}{4} + 2k\pi}{3} \quad k = 0, 1, 2 \]
05

- Find all solutions for argument (a)

Calculate the arguments for each value of k: \[ \theta_0 = \frac{\pi}{12} \] \[ \theta_1 = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{9\pi}{12} = \frac{3\pi}{4} \] \[ \theta_2 = \frac{\pi}{12} + \frac{4\pi}{3} = \frac{17\pi}{12} \]
06

- Write the solutions in rectangular form (a)

Convert back to rectangular form: \[ z_0 = 4 \left(\cos \frac{\pi}{12} + j \sin \frac{\pi}{12}\right) \] \[ z_1 = 4 \left(\cos \frac{3\pi}{4} + j \sin \frac{3\pi}{4}\right) \] \[ z_2 = 4 \left(\cos \frac{17\pi}{12} + j \sin \frac{17\pi}{12}\right) \]
07

- Rewrite the equation in standard form (b)

Rewriting the equation: \[ z^3 = 2 + 5j \]
08

- Express in polar form (b)

Convert to polar form: \[ 2 + 5j = r (\cos \theta + j \sin \theta) \] Find r and θ such that \[ r = \sqrt{2^2 + 5^2} = \sqrt{29} \] and \[ \tan \theta = \frac{5}{2} \Rightarrow \theta = \arctan \frac{5}{2} \]
09

- Use De Moivre's Theorem (b)

Using De Moivre’s theorem: \[ z = \sqrt[3]{\sqrt{29}} \left( \cos \frac{\theta + 2k\pi}{3} + j \sin \frac{\theta + 2k\pi}{3} \right) \] where \( k = 0, 1, 2 \)
10

- Find all solutions for argument (b)

Calculate arguments for each value of k: \[ \theta_0 = \frac{\arctan \frac{5}{2}}{3} \] \[ \theta_1 = \frac{\arctan \frac{5}{2} + 2\pi}{3} \] \[ \theta_2 = \frac{\arctan \frac{5}{2} + 4\pi}{3} \]
11

- Convert to rectangular form (b)

Convert back to rectangular form: \[ z_0 = \sqrt[3]{\sqrt{29}} \left(\cos \frac{\arctan \frac{5}{2}}{3} + j \sin \frac{\arctan \frac{5}{2}}{3}\right) \] \[ z_1 = \sqrt[3]{\sqrt{29}} \left(\cos \frac{\arctan \frac{5}{2} + 2\pi}{3} + j \sin \frac{\arctan \frac{5}{2} + 2\pi}{3}\right) \] \[ z_2 = \sqrt[3]{\sqrt{29}} \left(\cos \frac{\arctan \frac{5}{2} + 4\pi}{3} + j \sin \frac{\arctan \frac{5}{2} + 4\pi}{3}\right) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

De Moivre's Theorem
De Moivre's Theorem is a vital tool in working with complex numbers, especially when they're in polar form. This theorem states that for any complex number \(z = r(\text{cos} \theta + j \text{sin} \theta)\) and any integer n, the n-th power of z can be given by:\[z^n = r^n (\text{cos}(n\theta) + j \text{sin}(n\theta))\]. This helps simplify the process of raising complex numbers to a power. The theorem makes it easier by keeping the magnitude and multiplying the argument by the power.
Complex Numbers
Complex numbers extend the idea of real numbers by adding an imaginary part. They are written in the form \(a + bj\), where \(a\) and \(b\) are real numbers, and \(j\) is the imaginary unit (with the property that \(j^2 = -1\)). Complex numbers can be represented as points in a plane, known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Polar Coordinates
Polar coordinates provide an alternative way to represent complex numbers. Instead of using \(a + bj\), we use \(r(\text{cos} \theta + j \text{sin} \theta)\), where \(r\) is the magnitude (distance from the origin) and \(\theta\) is the argument (the angle with the positive x-axis). This form is useful in many contexts, including simplifying multiplication and division of complex numbers.
Argument of a Complex Number
The argument of a complex number is the angle it makes with the positive x-axis in the complex plane. It is often denoted by \(\theta\). To find the argument, you can use the arctangent function: \[\theta = \text{arctan} \frac{b}{a}\]. This angle can also be adjusted by adding integer multiples of \(2\pi\) to account for the periodicity of trigonometric functions.
Rectangular Form of Complex Numbers
The rectangular form of a complex number is simply \(a + bj\), where \(a\) is the real part and \(b\) is the imaginary part. This form is called 'rectangular' because each complex number corresponds to a point (a, b) in the complex plane. This form helps in performing basic arithmetic operations like addition and subtraction easily.

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Most popular questions from this chapter

Zeigen Sie: F?r jede komplexe Zahl \(z=x+j y\) gilt: a) \(z+z^{*}=2 \cdot \operatorname{Re}(z)\) b) \(z-z^{*}=2 \mathrm{j} \cdot \operatorname{Im}(z)\)

Bringen Sie die in der Polarform vorliegenden komplexen Zahlen $$ \begin{array}{ll} z_{1}=4(\cos 1+\mathrm{j} \cdot \sin 1), & z_{2}=3 \cdot \mathrm{e}^{\mathrm{j} 30^{\circ}}, \quad z_{3}=5 \cdot \mathrm{e}^{j 135^{\circ}} \\ z_{4}=5\left[\cos \left(-60^{\circ}\right)+\mathrm{j} \cdot \sin \left(-60^{\circ}\right)\right], & z_{5}=2 \cdot \mathrm{e}^{-\mathrm{j} \frac{3}{2} \pi}, \quad z_{6}=\mathrm{e}^{\mathrm{j} 240^{\circ}} \\ z_{7}=2\left(\cos 210^{\circ}+\mathrm{j} \cdot \sin 210^{\circ}\right), & z_{8}=\cos (-0,5)+\mathrm{j} \cdot \sin (-0,5) \end{array} $$ in die kartesische Form und bestimmen Sie die jeweilige konjugiert komplexe Zahl.

Zeichnen Sie die Ortskurven der folgenden komplexen Zeiger: a) \(z(t)=a \cdot \cos t+j \cdot b \cdot \sin t \quad(0 \leqslant t<2 \pi)\) b) \(\quad z(t)=2 \cdot \cos ^{2} t+j \cdot \sin (2 t) \quad\left(0 \leqslant t<\frac{\pi}{2}\right)\)

Die folgenden harmonischen Schwingungen sind zun?chst in der Simusform \(y=A-\sin (\omega t+\varphi)\) mit \(A>0,0 \leqslant \varphi<2 \pi\) anzugeben und anschlieBend rechnerisch und zeichnerisch durch komplexe Sinuszeiger darzustellen. Wie lauten die komplexen Amplituden (in der kartesischen Form)? a) \(\quad y_{1}=3 \cdot \sin \left(2 t+\frac{\pi}{3}\right)\) b) \(y_{2}=-3 \cdot \sin \left(4 t-\frac{\pi}{4}\right)\) c) \(\quad y_{3}=-4 \cdot \sin \left(2 t+\frac{\pi}{2}\right)\) d) \(y_{4}=5 \cdot \sin (\pi t-1,2)\)

Die in der kartesischen Form gegebenen komplexen Zahlen $$ \begin{array}{llll} z_{1}=2+\pi \mathrm{j} . & z_{2}=4,5-2,4 \mathrm{j}, & z_{3}=-3+5 \mathrm{j}, & z_{4}=-6 \\ z_{5}=-3-2 \mathrm{j}, & z_{6}=-1+\mathrm{j}, & z_{7}=-4 \mathrm{j}, & z_{8}=-3-\mathrm{j} \end{array} $$ sind in die Polarform umzurechnen. Wie lauten die konjugiert komplexen Zahlen?
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