Step by step
The question
Question: write 2 + 2i in polar form.
Polar form gives its distance from 0 and its angle.
1 Plot it
The complex plane: the real part goes across, the i part goes up.
2 + 2i is the point (2, 2): 2 right, 2 up.
r is its distance from 0.
θ is its angle, from the positive real axis, turning counterclockwise.
2 Find r
Pythagoras: r = √(a² + b²), with a = 2 and b = 2.
2 squared is 4.
2 squared is 4.
So r = 2√2.
3 Find θ
The reference angle: tan is up over across, without the signs.
That is the tan of 45°, from the unit circle.
The point is in quadrant I, so θ = 45°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So 2 + 2i = 2√2(cos 45° + i sin 45°).
5 Check
Let's check: back to a + bi.
2√2 times cos 45° is 2.
2√2 times sin 45° is 2.
That is 2 + 2i again.
Right!
Careful!
The angle is always measured from the positive real axis, turning counterclockwise.
Let's look back.
First: plot it.
Then: find r.
Then: find θ.
Then: write it in polar form.
Last: check.