/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Solve the equation \(z^{4}+1=\ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 13

Solve the equation \(z^{4}+1=\mathrm{j} \sqrt{3}\).

Short Answer

Expert verified
The four solutions are: \( z_0 = \sqrt{2} e^{j(-\frac{\pi}{12})}, z_1 = \sqrt{2} e^{j(\frac{\pi}{12})}, z_2 = \sqrt{2} e^{j(\frac{5\pi}{12})}, z_3 = \sqrt{2} e^{j(\frac{3\pi}{4})} \).

Step by step solution

01

Set Up the Complex Equation

The equation given is \( z^4 + 1 = j \sqrt{3} \). Rearrange it to isolate the term involving \( z \):\[ z^4 = j \sqrt{3} - 1 \]
02

Express in Polar Form

Convert the right side of the equation \( j \sqrt{3} - 1 \) into polar form. Start by finding the modulus:\[ |j \sqrt{3} - 1| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \]Find the argument \( \theta \) (angle with the positive x-axis):\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \]So in polar form, \( j \sqrt{3} - 1 = 2 \left( \cos(-\frac{\pi}{3}) + j \sin(-\frac{\pi}{3}) \right) \).
03

Solve Using De Moivre's Theorem

Using De Moivre's Theorem, express \( z^4 = 2 e^{j(-\frac{\pi}{3})} \). Thus:\[ z = 2^{1/4} e^{j\left(\frac{-\pi}{12} + \frac{k\pi}{2}\right)} \text{ for } k = 0, 1, 2, 3 \]
04

Calculate Four Different Solutions

Calculate for each \( k \):- For \( k = 0 \): \[ z_0 = \sqrt{2} e^{j(-\frac{\pi}{12})} = \sqrt{2} \left( \cos\left(-\frac{\pi}{12}\right) + j \sin\left(-\frac{\pi}{12}\right) \right) \]- For \( k = 1 \): \[ z_1 = \sqrt{2} e^{j\left( \frac{\pi}{12} \right)} = \sqrt{2} \left( \cos\left( \frac{\pi}{12} \right) + j \sin\left( \frac{\pi}{12} \right) \right) \]- For \( k = 2 \): \[ z_2 = \sqrt{2} e^{j\left( \frac{5\pi}{12} \right)} = \sqrt{2} \left( \cos\left( \frac{5\pi}{12} \right) + j \sin\left( \frac{5\pi}{12} \right) \right) \]- For \( k = 3 \): \[ z_3 = \sqrt{2} e^{j\left( \frac{9\pi}{12} \right)} = \sqrt{2} \left( \cos\left( \frac{3\pi}{4} \right) + j \sin\left( \frac{3\pi}{4} \right) \right) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polar Form
Complex numbers can be expressed in a polar form, which is a different way from the usual rectangular form. While the rectangular form uses real and imaginary components, polar form uses a modulus (magnitude) and an argument (angle).

To convert a complex number to polar form, follow these steps:
  • Find the modulus, which is the length of the vector in the complex plane. It is calculated as \( |a + bj| = \sqrt{a^2 + b^2} \).
  • Determine the argument, the angle formed with the positive x-axis, using the function \( \theta = \tan^{-1}(b/a) \).
  • Combine the modulus and argument: \( z = r (\cos(\theta) + j \sin(\theta)) \) or in exponential form, \( z = r e^{j\theta} \).
This transformation is crucial when you need to multiply or divide complex numbers, as operations in polar form are more straightforward than in rectangular form.
De Moivre's Theorem
De Moivre's Theorem provides a powerful method for finding powers and roots of complex numbers expressed in polar form. It states that for a complex number \( z = r (\cos(\theta) + j\sin(\theta)) \):
\[ z^n = r^n (\cos(n\theta) + j\sin(n\theta)) \]
This theorem simplifies many calculations because raising complex numbers to powers or extracting roots can be cumbersome in rectangular form. When solving equations like \( z^4 = 2 e^{j(-\frac{\pi}{3})} \), it becomes straightforward with De Moivre鈥檚 Theorem.

To apply it:
  • Identify \( r \) and \( \theta \) from the initial complex number.
  • Apply the theorem to get the expression for \( z^n \) or solve for roots by adjusting \( \theta \) accordingly.
This method allows for the computation of all possible solutions, considering the periodic nature of angle rotations around a circle.
Complex Modulus
The modulus of a complex number, often termed as the magnitude, is a measure of its size or distance from the origin in the complex plane. It helps to "quantify" the complex number, providing a basis for understanding its scale.

To compute the modulus for any complex number \( z = a + bj \), use the formula:
  • \( |z| = \sqrt{a^2 + b^2} \)
  • For example, if \( z = j \sqrt{3} - 1 \), then the modulus is \( |z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2 \).
Knowing how to calculate the modulus is critical because it appears frequently in polar coordinates and plays a major role in De Moivre's Theorem. It is particularly handy for understanding and predicting the behavior of complex numbers when they are visualized in the polar coordinate system.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sketch an Argand diagram and upon it mark the poles of the rational function $$ G(s)=\frac{3(s-3)}{(s+1)(s+1+3 \mathrm{j})(s+1-3 \mathrm{j})} $$

Consider the complex number $$ z=\cos \theta+\mathrm{j} \sin \theta $$ (a) Write down the exponential form of this number. (b) By raising the exponential form to the power \(n\), and using one of the laws of indices, deduce De Moivre's theorem.

In each case, draw an Argand diagram and shade the region that satisfies (a) \(\operatorname{Re}(z)>0\) (b) \(\operatorname{Re}(z) \leq-2\) (c) \(\operatorname{Im}(z)<3\) (d) \(\operatorname{Im}(z) \geq-3\) where \(\operatorname{Re}(z)\) stands for the real part of \(z\), and \(\operatorname{Im}(z)\) stands for imaginary part of \(z\).

Express \(z=-3+2 \mathrm{j}\) in polar form and hence find \(z^{6}\), converting your answer into cartesian form.

Use De Moivre's theorem to show that if \(z=\cos \theta+\mathrm{j} \sin \theta\) then (a) \(z^{\prime \prime}=\cos n \theta+\mathrm{j} \sin n \theta\) (b) \(z^{-n}=\cos n \theta-\mathrm{j} \sin n \theta\) Deduce that $$ z^{n}+\frac{1}{z^{n}}=2 \cos n \theta $$ and $$ z^{n}-\frac{1}{z^{n}}=2 \mathrm{j} \sin n \theta $$
See all solutions

Recommended explanations on Physics Textbooks

Oscillations

Read Explanation

Force

Read Explanation

Geometrical and Physical Optics

Read Explanation

Electricity and Magnetism

Read Explanation

Electric Charge Field and Potential

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.