Step by step
The question
Question: write √3 − 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.
√3 − i is the point (1.73, −1): 1.73 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 = √3 and b = −1.
√3 squared is 3.
−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 30°, from the unit circle.
The point is in quadrant IV: 360° − 30°.
So θ = 330°.
Careful!
The calculator gives −30°.
That is the same direction as 330°.
We keep θ from 0° to 360°.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So √3 − i = 2(cos 330° + i sin 330°).
5 Check
Let's check: back to a + bi.
2 times cos 330° is √3.
2 times sin 330° is −1.
That is √3 − i again.
Right!
Let's look back.
First: plot it.
Then: find r.
Then: find θ.
Then: write it in polar form.
Last: check.