I will give 55 points please help!!! It's super important Convert the rectangular form of the complex number 2-2i into polar form. Show all work and label the modulus and argument.

polar form = [tex]r(cos \theta + i sin \theta)[/tex]
r = modulus
theta = argument

[tex]r = \sqrt{a^2 + b^2} = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2 \sqrt{2}\\ \\ tan \theta = \frac{b}{a} = \frac{-2}{2} = -1 \\ \\ \theta = \frac{7 \pi}{4} [/tex]

Note: the point '2-2i' in the complex plane is in the 4th quadrant, therefore angle must be between 3pi/2 and 2pi.

Answer:
[tex]2-2i = 2 \sqrt{2} (cos \frac{7 \pi}{4} + i sin \frac{7 \pi}{4})[/tex]

shivishivangi1679 shivishivangi1679 · Answer 2 · 2022-04-13T09:13:28+02:00

The modulus and argument are 2√2 and 3π/4. The polar form is 2√2(cos 3π/4 + i sin 3π/4).

What is the polar form of a complex number?

The polar form of a complex number is a form of representing a complex number. It can be shown as Z = a + ib which is the rectangular form of a complex number.

Also, the polar form of a complex number with coordinate (x, y)

= r ( cos Ф + i sin Ф)

where r is a modulus, Ф is an argument

The given polar form is 2 - 2i

[tex]r = \sqrt{x^{2} +y^{2} } \\r = \sqrt{2^{2} +(-2)^{2} }\\r = 2\sqrt{2}[/tex]

tan Ф = y/a = -2/2

tan Ф = -1 = 3π/4

Therefore, the polar form is 2√2(cos 3π/4 + i sin 3π/4).

Learn more about polar form;

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