Expert: Alon Mandes - 2/25/2009

Question
how do i get standard form (a+bi) from this...

2(cos(pi/4)+isin(pi/4)

That is the polar form equation i got from dividing.. but since after canceling i end up with sqrt2-sqrt2i... it doesn't work...

another problem similar is:

2(cos(7pi/4)+isin(7pi/4))

I have the same situation as above.


~*~*~

also the division problems:

1. 4+4i
  -----  division  
  -1 +i

2.     -7i
  -----------  division
   sqrt(3)+ i




also i know this is a lot of questions to ask.. but i really cannot figure out the rules for finding the angles... like cos-1.... -cos-1... and the quadrants... any advice or tips???

Answer
The transfer rule from polar to Cartesian is :
Z=a+ib <--> Z=ρe^(iθ) Where :
ρ=√(a²+b²)
θ=Arctang(b/a)
OR
a=ρCos(θ)
b=ρSin(θ)
For example :
(in cartezian) : Z=2Cos(π/4)+iSin(π/4)=√2+i√2
(in polar)     : ρ=√(2+2)=2 & θ=Arctang(√2/√2)=π/4, so Z=2e^(iπ/4).

In division we may use 1 of 2 methods, which both yield the same.
Method 1 : We multiply the the fraction by the complex conjugate of the lower part of the fraction. Meaning :

4+4i
---- =
-1+i

(4+4i)(-1-i)
------------- =
(-1+i)(-1-i)

-4-4i-4i+4
---------- =
1+1

-8i
------- = -4i      
2

Method 2 : To transfer the Cartesian form into polar form & to
apply the usual exponential rules.

Alon.