# polar form, urgent

• Jan 13th 2008, 09:03 PM
Revolt
polar form, urgent
ok, exam in 2 hours and i dont need the answer to this question, i need a list of the steps i take in order to do it, would be hugely grateful.

z = 1 - i
w = 1+ i

a) write in polar form
b) evaluate z^3/w^7 in form a+ib
c) find all cube roots of z leaving answers in polar form

thanks if anyone can help
• Jan 13th 2008, 09:26 PM
Jhevon
don't you think this is cutting it a bit close. asking for help 2 hours before an exam?
Quote:

Originally Posted by Revolt
ok, exam in 2 hours and i dont need the answer to this question, i need a list of the steps i take in order to do it, would be hugely grateful.

z = 1 - i
w = 1+ i

a) write in polar form

the polar form of a complex number $\displaystyle a + ib$ is: $\displaystyle re^{i \theta}$

where $\displaystyle r = \sqrt{a^2 + b^2}$ and $\displaystyle \theta = \arctan \frac ba$ (usually)

Quote:

b) evaluate z^3/w^7 in form a+ib
find and simplify $\displaystyle \frac {z^3}{w^7}$ with $\displaystyle z$ and $\displaystyle w$ in their polar forms. then convert back to $\displaystyle a + ib$ form

Quote:

c) find all cube roots of z leaving answers in polar form
see post #4 here

of course, we would write $\displaystyle e^{i \theta}$ instead of $\displaystyle \cos \theta + i \sin \theta$ as was done there

try them and tell me what you get. it will be good practice for that exam of yours
• Jan 13th 2008, 09:31 PM
CaptainBlack
Quote:

Originally Posted by Revolt
ok, exam in 2 hours and i dont need the answer to this question, i need a list of the steps i take in order to do it, would be hugely grateful.

z = 1 - i
w = 1+ i

a) write in polar form
b) evaluate z^3/w^7 in form a+ib
c) find all cube roots of z leaving answers in polar form

thanks if anyone can help

(a)

$\displaystyle z=r e^{i \theta}=r \cos(\theta) + r \sin(\theta) = 1-i$

so:

$\displaystyle r=\sqrt{2}$ ,

and $\displaystyle \theta$ is the angle such that:

$\displaystyle \cos(\theta)=1/\sqrt(2)$

and

$\displaystyle \sin(\theta)=-1/\sqrt(2).$

A similar procedure will turn w into polar form: $\displaystyle w=s e^{i \phi}$

(b) $\displaystyle z^3/w^7 = (r^4/s^7)(e^{i 3 \theta}/e^{i 7 \phi})= (r^4/s^7)e^{i (3 \theta-7 \phi)}$

which you should be able to turn back into cartesian form.

(c) $\displaystyle z=r e^{i \theta}$ where $\displaystyle r$ and $\displaystyle \theta$ take the values found earlier.

Then we may write:

$\displaystyle z=r e^{i \theta+2 i n \pi},\ n\in \mathbb{Z}$ (an integer)

then

$\displaystyle z^{1/3}=r^{1/3} e^{(i \theta+2 i n \pi)/3},\ n\in \mathbb{Z}$

and there are just three distinct values for this corresponding to $\displaystyle n=0,1,2$

RonL
• Jan 13th 2008, 09:39 PM
noreaction
z = 1 - i
w = 1+ i

Quote:

a) write in polar form
Convert to polar form:

find the modulus $\displaystyle |z|$(also written as r), which is $\displaystyle \sqrt{(x^2 + y^2)}$
in the case of z $\displaystyle \sqrt{(1^2 + 1^2)}= \sqrt 2$
z = r = $\displaystyle \sqrt2$
Then find the argument $\displaystyle \theta$. Since $\displaystyle tan \theta = \frac {rise}{run}$ you can easily find the argument here.

Polar form would be $\displaystyle r (cos \theta + isin \theta)$

Quote:

b) evaluate z^3/w^7 in form a+ib
To find exponents of complex numbers, use De Moivre's theorem

If $\displaystyle z = r (cos \theta + isin \theta)$
then $\displaystyle z^n = r^n (cos (n\theta) + isin (n\theta))$

After this, convert back to a+ib form by multiplying r to $\displaystyle (cos \theta + isin \theta)$

Quote:

c) find all cube roots of z leaving answers in polar form
Assume $\displaystyle z = m^3$ where m is a cube root of z

so $\displaystyle m = \sqrt[3]r (cos \frac {\theta}{3} + i sin \frac {\theta}{3})$
• Jan 13th 2008, 09:42 PM
CaptainBlack
Quote:

Originally Posted by Jhevon

where $\displaystyle r = \sqrt{a^2 + b^2}$ and $\displaystyle \theta = \arctan \frac ba$ (usually)

$\displaystyle \arctan$ does not do here as we need the angle in the range $\displaystyle [0 , 2 \pi)$ (or equivalent) but $\displaystyle \arctan$ gives an answer in the range $\displaystyle (\pi/2,\pi/2]$ (or equivalent).

RonL
• Jan 13th 2008, 09:44 PM
CaptainBlack
Quote:

Originally Posted by noreaction
c) find all cube roots of z leaving answers in polar form

Assume $\displaystyle z = m^3$ where m is a cube root of z

so $\displaystyle m = \sqrt[3]r (cos \frac {\theta}{3} + i sin \frac {\theta}{3})$

How many cube roots of $\displaystyle z$ does that give us? Should there not be 3?

RonL
• Jan 13th 2008, 09:47 PM
noreaction
Sorry, i forgot to put $\displaystyle +2k\pi$ for the angles