Convert the rectangular form of the complex number 2−2i into polar form. Show all work and label the modulus and argument.
To convert the complex number 2−2i from rectangular form to polar form, we first need to find the modulus and the argument.
Given complex number: z = 2−2i
Modulus (r):
r = √(2^2 + (-2)^2)
r = √(4 + 4)
r = √8
r = 2√2
Argument (θ):
θ = arctan(-2/2)
θ = arctan(-1)
θ = -π/4
Now, we can express the complex number in polar form:
z = 2√2*(cos(-π/4) + i*sin(-π/4))
Therefore, the polar form of the complex number 2−2i is:
z = 2√2*(cos(-π/4) + i*sin(-π/4))