Step by step
The question
Question: write 1 − i 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.
1 − i is the point (1, −1): 1 right, 1 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 = 1 and b = −1.
1 squared is 1.
−1 squared is 1.
So r = √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 IV: 360° − 45°.
So θ = 315°.
Careful!
The calculator gives −45°.
That is the same direction as 315°.
We keep θ from 0° to 360°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So 1 − i = √2(cos 315° + i sin 315°).
5 Check
Let's check: back to a + bi.
√2 times cos 315° is 1.
√2 times sin 315° is −1.
That is 1 − i again.
Right!
Let's look back.
First: plot it.
Then: find r.
Then: find θ.
Then: write it in polar form.
Last: check.