/*! 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 26 Plot each complex number. Then w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 26

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$1-i \sqrt{5}$$

Short Answer

Expert verified
The polar form of the complex number \(1 - i \sqrt{5}\) is \(\sqrt{6}[\cos(-\arctan(\sqrt{5})) + i \sin(-\arctan(\sqrt{5}))]\).

Step by step solution

01

Understanding the Complex Number

The complex number given is \(1 - i \sqrt{5}\). Here, the real part is 1, and the imaginary part is \(-\sqrt{5}\).
02

Plotting the complex number

Plot the complex number on the complex plane using the real part of the complex number as the x-coordinate and the imaginary part as the y-coordinate. That is, plot the point \(1, -\sqrt{5}\) on the complex plane.
03

Calculating the magnitude (r)

To calculate the magnitude of the vector representing the complex number, apply the Pythagorean theorem. \nThe formula is \(r = \sqrt{a^2 + b^2}\), where 'a' is the real part and 'b' is the imaginary part. Thus, \(r = \sqrt{1^2 + (-\sqrt{5})^2} = \sqrt{1+5} = \sqrt{6}\).
04

Calculating the argument (胃)

The argument of a complex number is the angle it makes with the positive X-axis. It is calculated using the formula \(胃 = \arctan(\frac{b}{a})\), where 'a' is the real part of the complex number and 'b' is the imaginary part. Given that 'a' is 1 and 'b' is \(-\sqrt{5}\), \(胃 = \arctan(\frac{-\sqrt{5}}{1}) = -\arctan(\sqrt{5})\). This value can be in degrees or radians, depending on what you decide.
05

Writing the Complex Number in Polar Form

Finally, write this complex number in polar form using the found r and 胃. The polar form of the complex number is: \(r (\cos(胃) + i \sin(胃))\). Here, \(r = \sqrt{6}\) and 胃 is \(-\arctan( \sqrt{5})\), so the final polar form will be: \(\sqrt{6}[\cos(-\arctan(\sqrt{5})) + i \sin(-\arctan(\sqrt{5}))]\).

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

If you are given two sides of a triangle and their included angle, you can find the triangle's area. Can the Law of Sines be used to solve the triangle with this given information? Explain your answer.

Write an equation in point-slope form and general form for the line passing through (-2,5) and perpendicular to the line whose equation is \(x-4 y+8=0\) (Section \(1.5,\) Example 2 )

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}$$

Use a graphing utility to graph \(r=1+2 \sin n \theta\) for \(n=1,2,3,4,5,\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of large and small petals that occur corresponding to each value of \(n\) ? How are the large and small petals related when \(n\) is odd and when \(n\) is even?

A wagon is pulled along level ground by exerting a force of 40 pounds on a handle that makes an angle of \(32^{\circ}\) with the horizontal. How much work is done pulling the wagon 100 feet? Round to the nearest foot-pound.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.