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 left, 2 down.
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 III: 180° + 45°.
So θ = 225°.
Careful!
The calculator's tan⁻¹ only gives angles in quadrants I and IV.
Our point is in quadrant III: look at the picture.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So −2 − 2i = 2√2(cos 225° + i sin 225°).
5 Check
Let's check: back to a + bi.
2√2 times cos 225° is −2.
2√2 times sin 225° is −2.
That is −2 − 2i again.
Right!
Let's look back.
First: plot it.
Then: find r.
Then: find θ.
Then: write it in polar form.
Last: check.