Step by step
The question
Question: write −1 + √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.
−1 + √3 i is the point (−1, 1.73): 1 left, 1.73 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 = −1 and b = √3.
−1 squared is 1.
√3 squared is 3.
So r = 2.
3 Find θ
The reference angle: tan is up over across, without the signs.
That is the tan of 60°, from the unit circle.
The point is in quadrant II: 180° − 60°.
So θ = 120°.
Careful!
The calculator's tan⁻¹ only gives angles in quadrants I and IV.
Our point is in quadrant II: look at the picture.
4 Write it in polar form
Polar form is r(cos θ + i sin θ).
So −1 + √3 i = 2(cos 120° + i sin 120°).
5 Check
Let's check: back to a + bi.
2 times cos 120° is −1.
2 times sin 120° is √3.
That is −1 + √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.