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 left, 1 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 = 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 II: 180° − 45°.
So θ = 135°.
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 + i = √2(cos 135° + i sin 135°).
5 Check
Let's check: back to a + bi.
√2 times cos 135° is −1.
√2 times sin 135° 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.