/*! 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 58 Use DeMoivre's Theorem to find t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 58

Use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$\left[\frac{1}{2}\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)\right]^{5}$$

Short Answer

Expert verified
The result of the given complex number raised to the fifth power is \(i\), or equivalently, \(0 + i*1\).

Step by step solution

01

Apply DeMoivre's Theorem to compute the power

DeMoivre's theorem states that \((cos(x) + i*sin(x))^n = cos(nx) + i*sin(nx)\). So \[\left(\frac{1}{2}\left(\cos \frac{\pi}{10}+i \sin\frac{\pi}{10}\right)\right)^{5} = \left(\frac{1}{2}\right)^5\left[\cos \left(5*\frac{\pi}{10}\right) + i \sin \left(5*\frac{\pi}{10}\right)\right] =\frac{1}{32}\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right)\]
02

Convert trigonometric form into rectangular form

The rectangular form of a complex number is \(a + i*b\), where \(a = R \cos x\), \(b = R \sin x\), and \(R\) is the radius. Here, \(R = \frac{1}{32}\), \(x = \frac{\pi}{2}\). Therefore, applying the definitions of cosine and sine for \(x = \frac{\pi}{2}\), we find that \(a = 0\) and \(b = 1\). So, the rectangular form is \(0 + i*1 = i\).
03

Write the final answer

The final answer is simply \(i\) or in terms of \(a + i*b\), it's \(0 + i*1\).

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!

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

Solve the equation \(2 x^{3}+5 x^{2}-4 x-3=0\) given that -3 is a zero of \(f(x)=2 x^{3}+5 x^{2}-4 x-3\) (Section \(2.4,\) Example 6 )

The components of \(\mathbf{v}=180 \mathbf{i}+450 \mathbf{j}\) represent the respective number of one-day and three-day videos rented from a video store. The components of \(\mathbf{w}=3 \mathbf{i}+2 \mathbf{j}\) represent the prices to rent the one-day and three-day videos, respectively. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}+8 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

How do you determine the work done by a force F in moving an object from \(A\) to \(B\) when the direction of the force is not along the line of motion?

Using words and no symbols, describe how to find the \(\mathrm{d}\) product of two vectors with the alternative formula $$\mathbf{v} \cdot \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \cos \theta$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

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.