Solved

Why can z^3 = -27 be written as r^3(cos(3(theta)+i.sin(3(theta))?

Posted on 2011-02-21
4
477 Views
Last Modified: 2012-05-11
I'm working with complex numbers and reading through a solution,

It begins like this - In polar form, -27 = 27 (cos(pi) + i sin(pi))

I get that, but then it says

If z = r(cos(theta)+i sin(theta), then the equation z^3 = -27 can be written as

r^3(cos(3(theta)+i.sin(3(theta)) = 27(cos(pi) + i sin(pi))

I don't see where the left hand side came from??


0
Comment
Question by:purplesoup
  • 2
4 Comments
 
LVL 32

Expert Comment

by:phoffric
ID: 34942649
It comes from Euler's formula - http://en.wikipedia.org/wiki/Euler's_formula

e^(i theta) == cos(theta) + i sin(theta)

0
 
LVL 32

Accepted Solution

by:
phoffric earned 250 total points
ID: 34942657
z  = r( cos(theta) + i sin(theta) )
 z³ = r³( cos(theta) + i sin(theta) )³

Convert to e^p
raise to the third power
and convert back again to polar form
0
 
LVL 18

Assisted Solution

by:deighton
deighton earned 250 total points
ID: 34942760
lets see

z1 = r1(cos(A)+i sin(A))
z2 = r2(cos(B)+i sin(B))

z1 x z2 = r1(cos(A)+i sin(A)) r2(cos(B)+i sin(B))

= r1 r2 [cos(A) cos(B) + i sin(A) cos(B) + i cos(A) sin(B) - sin(A) sin(B)]

now since we know where we want to go, let us recall the double angle identities

sin(A + B) = sin(A)cos(B) + sin(B) cos(A)
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)


using these

 r1 r2 [cos(A) cos(B) + i sin(A) cos(B) + i cos(A) sin(B) - sin(A) sin(B)]

=  r1 r2 [cos(A) cos(B)  - sin(A) sin(B) +  i sin(A) cos(B) + i cos(A) sin(B))]

=  r1 r2 [cos(A+B) +  i sin(A + B)]

QED

so in general, the argument angle is always added when multiplying two complex numbers.  So if a number is cubed, the angle is tripled

if the angle exceeds 360 degrees (or 2 pi radians as you will have to come to know it) the argument can be reduced back to the corresponding value 0<= arg <= 2pi








0
 

Author Closing Comment

by:purplesoup
ID: 34942901
That's great - thanks!
0

Featured Post

Is Your Active Directory as Secure as You Think?

More than 75% of all records are compromised because of the loss or theft of a privileged credential. Experts have been exploring Active Directory infrastructure to identify key threats and establish best practices for keeping data safe. Attend this month’s webinar to learn more.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
technology for 'real' windows (not OS) 7 60
Cumulative Distribution 2 34
Logarithms 2 60
Vertex form of the function 8 66
Foreword (May 2015) This web page has appeared at Google.  It's definitely worth considering! https://www.google.com/about/careers/students/guide-to-technical-development.html How to Know You are Making a Difference at EE In August, 2013, one …
Lithium-ion batteries area cornerstone of today's portable electronic devices, and even though they are relied upon heavily, their chemistry and origin are not of common knowledge. This article is about a device on which every smartphone, laptop, an…
Internet Business Fax to Email Made Easy - With  eFax Corporate (http://www.enterprise.efax.com), you'll receive a dedicated online fax number, which is used the same way as a typical analog fax number. You'll receive secure faxes in your email, f…
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.

896 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question

Need Help in Real-Time?

Connect with top rated Experts

11 Experts available now in Live!

Get 1:1 Help Now