Write -√3 + i in polar form Step by Step

Step by step

The question

Question: write −√3 + i in polar form.

Polar form gives its distance from 0 and its angle.

−√3 + i

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 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 = −√3 and b = 1.

r = √(−√3)² + 1²

−√3 squared is 3.

1 squared is 1.

So r = 2.

r = √3 + 1 = √4 = 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.

tan (ref) = 1√3 → 30°

The point is in quadrant II: 180° − 30°.

So θ = 150°.

θ = 180° − 30° = 150°

Careful!

The calculator's tan⁻¹ only gives angles in quadrants I and IV.

Our point is in quadrant II: look at the picture.

Careful: not θ = tan⁻¹(b ÷ a) = −30° ✗

4 Write it in polar form

Polar form is r(cos θ + i sin θ).

So −√3 + i = 2(cos 150° + i sin 150°).

= −√3 + i = 2(cos 150° + i sin 150°)

5 Check

Let's check: back to a + bi.

2 times cos 150° is −√3.

2 × cos 150° = 2 × (−√32) = −√3

2 times sin 150° is 1.

That is −√3 + i again.

Right!

2 × sin 150° = 2 × 12 = 1

Let's look back.

First: plot it.

Then: find r.

Then: find θ.

Then: write it in polar form.

Last: check.

Answer