Is the Complex Number z on a Circle?

[scoutfai]

The question is :
The complex number $$z$$ is given by
z = 1 + cos (theta) + i*sin (theta)
where -Pi < theta <= Pi
Show that for all values of theta, the point representing z in a Argrand Diagram is located on a circle. Find the centre and radius of the circle.

Note that i understand perhaps some of you don't understand the symbol i used above, hence i will explain it here :
smaller or equal to : <=
theta : an angle
Pi : a usefull constant in circle, i don't know how to define it.
i = square root of -1

 

[arildno]

Hint: go into the the reference frame with translated origin at 1+0i.

 

[matt grime]

I think we all new what the symbols meant, if not we'd probably not be able to help...

do you know the general formula for a circle in the complex plane? It's the set S ={ z | |z-w|=r} for some fixed w in the plane and r a positive constant. Do any choices of w and r spring to mind given we know that cos^2+sin^2 is identically 1?

Of course there are other ways to see the answer since it is translation of the unit circle (centred on the origin).

 

[poolwin2001]

Solution

Let me denote
: theta as x
:z* the complex conjugate of z
:^ denotes power x^2 ==x squared



then
Z=1+Cosx+iSinx
Z*=1+Cosx-iSinx

ZZ*=2 + 2Cosx^2 //z multiplied by (z*) will give you that

ZZ*-Z-Z*=
=2 + 2Cosx^2 - 1 - Cosx - iSinx - 1 - Cosx + iSinx
=2

therefore
ZZ* - Z - Z* + 1=3

==>Z(Z*-1) - (Z*-1) = 3
==>(Z-1)(Z*-1) = 3
==>(Z-1)(Z-1)*=3
==>|Z-1|^2=3
==>|Z-1|=real
Which is the general form of a circle in complex plane as matt grime pointed out.

Hope you found this solution useful :shy:

cheers
poolwin2001

P.S:Where did you get this question ??

 

[flexten]

This any help ?

$\mbox{\Huge \[ e^{i\theta } = cos\theta + isin\theta \]}$

Best

 

[scoutfai]

hard to understand

First of all, thanks to those people who had post reply to my question, your message is meaningful to me, thanks you.
Well, this question i get it from my work book in complex number exercise. Note that i am not European people, i live in South East Asian, hence i don't think is meaning to you if i told you which book i found this question, ok poolwin2001 ?

matt grime, i am just begin the Complex Number chapter, hence i really don't know got such Set which define any circle in the Argand Diagram. And to poolwin2001 also, i understand all your working but just don't understand how can u state that once u able to show |Z-1|=real , this implies that the point of complex number Z is located on a circle in Argand Diagram.

I just begin study this chapter, hence i don't think that the author expect me to use this kind of "theorem" ( which i think is under Further Mathematic ) to show Z is located in a circle, do you have any other easier method ?
For your information, i just study the operation of Complex Number (plus, minus, multiply and divide), conjugate ,draw Argand Diagram, that is all in my syllabus, and that question i was found it inside my textbook (so i sure the author will expect us use the method i study to solve it, and not using those "theorem" )

 

[matt grime]

The idea that these points define a circle doesn't have anything to do with the complex numbers as such.

You should be thinking of the numbers as points in the complex plane, yes? then | | means distance. so {z| |z-w|=r} means all the points z a distance r from w. This is just geometry in the plane, but with nicer notation.

 

[Leong]

what matt grime probably asked you to do was to use the trigonometry identity.

Let z= x + iy
=$$(1+cos\theta) + isin\theta$$

$$\Rightarrow x = 1+cos\theta$$
$$\Rightarrow y = sin\theta$$

$$sin^2\theta + cos^2\theta = 1$$
$$y^2 + (x-1)^2 = 1$$

Compare with the circle equation : $$(x-a)^2 + (y-b)^2 = r^2$$
with center coordinate of (a,b) and radius r.

 

[Ethereal]

I think I got the explanation down in the zip file (couldn't upload the image directly). Hope it helps, sure took me a long time to create it.

 

[poolwin2001]

wow 3 different solutions

WOW 3 different solutions
Great !3 different solns each radically different have been given .Awesome!
(But mine had to be the longest )