/*! 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 31 Write each complex number in rec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 31

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$8\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)$$

Short Answer

Expert verified
The complex number in rectangular form is \(5.7 - 5.7i\).

Step by step solution

01

Identify the magnitude and angle

The magnitude \(r\) and angle \(\theta\) of the complex number in polar form are given as \(r = 8\) and \(\theta = \frac{7 \pi}{4}\).
02

Calculate the real part

The real part of the complex number \(x\) in rectangular form is calculated by multiplying the magnitude \(r\) by the cosine of the angle \(\theta\). This gives us \(x = r \cos \theta = 8 \cos {\frac{7 \pi}{4}}\). Therefore, the real part \(x\) of the complex number is approximately 5.7 after rounding to the nearest tenth.
03

Calculate the imaginary part

The imaginary part of the complex number \(y\) in rectangular form is calculated by multiplying the magnitude \(r\) by the sine of the angle \(\theta\). Therefore, \(y = r \sin \theta = 8 \sin {\frac{7 \pi}{4}}\). Hence, the imaginary part \(y\) of the complex number is approximately -5.7 after rounding to the nearest tenth.
04

Write the complex number in rectangular form

In rectangular form, a complex number is written as \(x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. Therefore, the complex number in rectangular form is \(5.7 - 5.7i\).

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

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{4} 4 \theta+\cos 3 \theta$$

Graph the spiral \(r=\frac{1}{\theta} .\) Use a [-1.6,1.6,1] by [-1,1,1] viewing rectangle. Let \(\theta \min =0\) and \(\theta \max =2 \pi,\) then \(\theta \min =0\) and \(\theta \max =4 \pi,\) and finally \(\theta \min =0\) and \(\theta \max =8 \pi\)

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+10 \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{i}$$

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$\begin{aligned}&r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15} \quad \text { (Use } \quad \theta \min =0 \quad \text { and }\\\ &\theta \max =20 \pi .)\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

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.