Problem 31 Use de Moivre's Theorem to find ... [FREE SOLUTION]

91影视

Trigonometry

Charles P. McKeague, Mark D. Turner

$Math Studyset 91影视 Explanations$ Math

8 Edition

Chapter 8: Problem 31

Use de Moivre's Theorem to find each of the following. Write your answer in standard form. $$ (1+i)^{4} $$

Short Answer

Expert verified

The result is $-4$.

Step by step solution

Convert to Polar Form

The complex number $1+i$ can be written in polar form as $r(\cos \theta + i \sin \theta)$. First, find the modulus $r$ using the formula $r = \sqrt{x^2 + y^2}$, where $x=1$ and $y=1$. This gives us $r = \sqrt{2}$. The argument $\theta$ is given by $\theta = \tan^{-1}(\frac{y}{x})$, which in this case is $\theta = \tan^{-1}(1) = \frac{\pi}{4}$. Therefore, the polar form is $\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$.

Apply de Moivre's Theorem

De Moivre's Theorem states that $[r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta))$. Apply the theorem using $r = \sqrt{2}$, $\theta = \frac{\pi}{4}$, and $n=4$. Calculate $r^n = (\sqrt{2})^4 = 4$. For the angle, calculate $n\theta = 4 \cdot \frac{\pi}{4} = \pi$. Thus, $(1+i)^4 = 4(\cos \pi + i \sin \pi)$.

Simplify Using Trigonometric Values

Substitute the trigonometric values: $\cos \pi = -1$ and $\sin \pi = 0$. Therefore, $4(\cos \pi + i \sin \pi) = 4(-1 + i\cdot 0) = -4$.

Write in Standard Form

The standard form of a complex number is $a + bi$, where $a$ and $b$ are real numbers. From the simplification, we have $-4 + 0i$. Thus, the final answer in standard form is $-4$.

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 of Complex Numbers

The polar form is a fascinating way to express complex numbers, using their magnitude and angle rather than just their real and imaginary parts. Instead of writing a complex number like this:

Standard form:
1 + i

The polar form expresses this number in terms of its distance from the origin (the modulus) and its direction (the argument or angle).

Polar form:
r(\cos \theta + i \sin \theta)

To convert a complex number to polar form, calculate:

The modulus, $r$, using the formula: $r = \sqrt{x^2 + y^2}$, where $x$ and $y$ are the real and imaginary components.
The argument, $\theta$, which is how much you need to rotate the angle from the positive x-axis, using $\theta = \tan^{-1}(\frac{y}{x})$.

Once these are found, you can easily substitute them into the polar form. It's an alternative representation useful for multiplying and dividing complex numbers, making operations much simpler.

Modulus and Argument of Complex Numbers

Understanding the modulus and the argument is crucial in working with complex numbers.

The modulus, $r$, tells you how far the complex number is from the origin in the Cartesian plane. It's like finding the hypotenuse of a right triangle!
In our case, for the number $1 + i$, the modulus is $\sqrt{2}$.
The argument, $\theta$, is the angle the line representing the complex number makes with the positive x-axis. It guides you in the direction you must turn from the positive x-axis to point towards the complex number in the plane.
For $1 + i$, the argument is $\frac{\pi}{4}$, or 45 degrees.

By using both the modulus and the argument, you can graphically interpret the complex number's position and understand its behavior during various operations.

Complex Number Operations

Many students find operations on complex numbers challenging when only using standard form. However, using polar form makes things much easier. Once you have a complex number in polar form, performing operations like multiplication and division can be very straightforward.
For complex number powers, De Moivre's Theorem is a handy tool. It states that if you have a complex number in polar form:

$[r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta))$

Using De Moivre's Theorem:

Raise the modulus to the power $n$. For example, $(\sqrt{2})^4 = 4$.
Multiply the argument by $n$: $n\cdot\frac{\pi}{4} = \pi$.

By converting complex number operations into polar form operations, you simplify them significantly, enjoying the elegance and simplicity that mathematics offers in its various forms.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use de Moivre's Theorem to find each of the following. Write your answer in standard form. $$ (1+i)^{4} $$

Short Answer

Step by step solution

Convert to Polar Form

Apply de Moivre's Theorem

Simplify Using Trigonometric Values

Write in Standard Form

Key Concepts

Polar Form of Complex Numbers

Modulus and Argument of Complex Numbers

Complex Number Operations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.