This is a free lesson from our course in Trigonometry
This lesson will help you to understand the concept of complex numbers in polar form and how to write complex numbers
in terms of a distance from the origin and a direction (or angle) from
the positive horizontal axis. It is explained by the instructor with the help of
video, how you can find the real (horizontal) and imaginary (vertical) components
in terms of r (the length of the vector) and
(the angle made with the real axis).Let's take case of x + iy. Using the relationships
x = r cos
, and y =
r sin
and r =
(x^{2}
+ y^{2}), the values of r and
can be obtained and then replace x and y in z = x
+ iy, which results in polar form of complex number
(Continued from above) where r is the absolute value (or modulus) and
is the argument of the complex number and is equal to tan^{1}(y/x).
Polar form is sometimes called the trigonometric form as well. The polar form of
a complex number is especially useful when you're working with powers and roots
of a complex number.
For example, to write the complex number (rectangular form) z = 1 + i into
polar form, first you compare it with z = x + yi and
see that z = 1 + i has rectangular coordinates (1, 1). Now find the values
of r and
and then plug in r cos
for x and r sin
for y in z = x + yi and factor out r.
It will yield to
2(cos
/4 + i sin
/4).
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