The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r
and θ
where r
is the length of the vector and θ
is the angle made with the real axis.
From Pythagorean Theorem :
r2=a2+b2
By using the basic trigonometric ratios :
cosθ=ar
and sinθ=br
.
Multiplying each side by r
:
rcosθ=a and rsinθ=b
The rectangular form of a complex number is given by
z=a+bi
.
Substitute the values of a
and b
.
z=a+bi =rcosθ+(rsinθ)i =r(cosθ+isinθ)
In the case of a complex number, r
represents the absolute value or modulus and the angle θ
is called the argument of the complex number.
This can be summarized as follows:
The polar form of a complex number z=a+bi
is z=r(cosθ+isinθ)
, where r=|z|=a2+b2−−−−−−√
, a=rcosθ and b=rsinθ
, and θ=tan−1(ba)
for a>0
and θ=tan−1(ba)+π
or θ=tan−1(ba)+180°
for a<0
.
Example:
Express the complex number in polar form.
5+2i
The polar form of a complex number z=a+bi
is z=r(cosθ+isinθ)
.
So, first find the absolute value of r
.
r=|z|=a2+b2−−−−−−√ =52+22−−−−−−√ =25+4−−−−−√ =29−−√ ≈5.39
Now find the argument θ
.
Since a>0
, use the formula θ=tan−1(ba)
.
θ=tan−1(25) ≈0.38
Note that here θ
is measured in radians.
Therefore, the polar form of 5+2i
is about 5.39(cos(0.38)+isin(0.38))
.