Complex Analysis- Lecture 7- Polar Coordinates

1.7 Polar Coordinates:-
If P is a point in the complex plan corresponding to the complex number (x,y) or Z=x+iy , Then we see from fig. that x=r cos??? ?, y=r sin?? where
r=?(x^2+y^2 )=|x+iy|=|Z|
is called the modulus or absolute value of Z , and ? , called the argument of Z=x+iy [denoted by arg (Z) ] ,
is the angle which line (OP) ? make with
the positive x axis. That is ?=arg(Z)
It follows that
Z=x+iy=r cos???+i r? sin??
=r(cos??+i sin?? )
which is called the polar form of complex number, r and ? are called polar coordinates.
For any complex number Z?0, there corresponds only one value of ? in 0???2? . However , any other interval of length 2? , for example -?The angle ? may be obtained from the formula ?=tan^(-1)??y/x? , that is arg(Z)=tan^(-1)??y/x? .

? sin?? cos?? tan??
?30?^° 1/2 ?3/2 1/?3
?60?^° ?3/2 1/2 ?3
?45?^° 1/?2 1/?2 1

?37?^° 3/5 4/5 3/4
?53?^° 4/5 3/5 4/3
0^° 0 1 0
?90?^° 1 0 ?
?180?^° 0 1 0
?270?^° -1 0 -?
?360?^° 0 1 0

sin??-??=-sin??
cos??-??=cos??
tan??-??=-tan??



EX:-Express each of complex numbers in polar form .
2+2?3 i
r=?(4+12)=4 ?=tan^(-1)??(2?3)/2? =60=?/3
Thus 2+2?3 i=4(cos???/3?+i sin???/3? )
-5+5i
r=?(5^2+5^2 )=5?2
?=tan^(-1)??5/(-5)?=-45=-?/4
Thus -5+5i=5?2 (cos?(-?/4)+i sin?(-?/4) )=5?2 (cos???/4?-i sin???/4? )
-3i H.W