Steply

Polar form of complex numbers, De Moivre's theorem

Precalculus · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Plot it
  2. Find r
  3. Find θ
  4. Write it in polar form
  5. Use De Moivre
  6. Work it out
  7. Back to a + bi
  8. Check

The steps for Simplify (1 + i)^8. Every solution names its steps, so you always know what comes next.

Worked examples

Write 1 + i in polar form Easy

The question

Question: write 1 + i in polar form.

Polar form gives its distance from 0 and its angle.

1 + i

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 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.

r = √12 + 12

1 squared is 1.

1 squared is 1.

So r = √2.

r = √1 + 1 = √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.

tan (ref) = 11 → 45°

The point is in quadrant I, so θ = 45°.

θ = 45°

4 Write it in polar form

Polar form is r(cos θ + i sin θ).

So 1 + i = √2(cos 45° + i sin 45°).

= 1 + i = √2(cos 45° + i sin 45°)

5 Check

Let's check: back to a + bi.

√2 times cos 45° is 1.

√2 × cos 45° = √2 × √22 = 1

√2 times sin 45° is 1.

That is 1 + i again.

Right!

√2 × sin 45° = √2 × √22 = 1

Careful!

The angle is always measured from the positive real axis, turning counterclockwise.

Careful: θ from the positive real axis

Let's look back.

First: plot it.

Then: find r.

Then: find θ.

Then: write it in polar form.

Last: check.

Answer
= 1 + i = √2(cos 45° + i sin 45°)

▶ Watch it solved

Write -1 + i in polar form Medium

The question

Question: write −1 + i in polar form.

Polar form gives its distance from 0 and its angle.

−1 + i

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.

r = √(−1)2 + 12

−1 squared is 1.

1 squared is 1.

So r = √2.

r = √1 + 1 = √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.

tan (ref) = 11 → 45°

The point is in quadrant II: 180° − 45°.

So θ = 135°.

θ = 180° − 45° = 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.

Careful: not θ = tan⁻¹(b ÷ a) = −45° ✗

4 Write it in polar form

Polar form is r(cos θ + i sin θ).

So −1 + i = √2(cos 135° + i sin 135°).

= −1 + i = √2(cos 135° + i sin 135°)

5 Check

Let's check: back to a + bi.

√2 times cos 135° is −1.

√2 × cos 135° = √2 × (−√22) = −1

√2 times sin 135° is 1.

That is −1 + i again.

Right!

√2 × sin 135° = √2 × √22 = 1

Let's look back.

First: plot it.

Then: find r.

Then: find θ.

Then: write it in polar form.

Last: check.

Answer
= −1 + i = √2(cos 135° + i sin 135°)

▶ Watch it solved

Simplify (1 + i)^8 Harder

The question

Question: work out (1 + i) to the power 8.

Multiplying it out would take forever.

(1 + i)8 =

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 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.

r = √12 + 12

1 squared is 1.

1 squared is 1.

So r = √2.

r = √1 + 1 = √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.

tan (ref) = 11 → 45°

The point is in quadrant I, so θ = 45°.

θ = 45°

4 Write it in polar form

Polar form is r(cos θ + i sin θ).

So 1 + i = √2(cos 45° + i sin 45°).

= 1 + i = √2(cos 45° + i sin 45°)

5 Use De Moivre

De Moivre's rule: to raise it to the power n, raise r to the power n, and multiply the angle by n.

[r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)

So r becomes √2 to the power 8, and the angle 8 times 45°.

(1 + i)8 = (√2)8(cos(8 × 45°) + i sin(8 × 45°))

6 Work it out

Careful!

r to the power 8 means 8 of them multiplied: not times 8.

Careful: not √2 × 8 = 8√2 ✗

√2 times √2 is 2.

8 of them is 4 such pairs.

So r to the 8 is 16.

(√2)8 = 16

8 times 45° is 360°.

Full turns change nothing: take away 360°.

0° points the same way.

8 × 45° = 360°, 360° − 360° = 0°

So we get 16(cos 0° + i sin 0°).

= 16(cos 0° + i sin 0°)

7 Back to a + bi

From the unit circle: cos 0° is 1, and sin 0° is 0.

= 16(1 + 0i)

Multiply by 16: 16.

(1 + i)8 = 16

8 Check

Let's check the size of our answer, by Pythagoras: 16.

size = √256 + 0 = 16

It must be r to the power 8: 16.

The same!

So it is right.

(√2)8 = 16

Let's look back.

First: plot it.

Then: find r.

Then: find θ.

Then: write it in polar form.

Then: use De Moivre.

Then: work it out.

Then: back to a + bi.

Last: check.

Answer
(1 + i)8 = 16

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. write 1 + i in polar form
    Answer
    = 1 + i = √2(cos 45° + i sin 45°)
  2. write √3 + i in polar form
    Answer
    = √3 + i = 2(cos 30° + i sin 30°)
  3. write 1 + i√3 in polar form
    Answer
    = 1 + √3 i = 2(cos 60° + i sin 60°)
  4. write 2 + 2i in polar form
    Answer
    = 2 + 2i = 2√2(cos 45° + i sin 45°)
  5. write 2√3 + 2i in polar form
    Answer
    = 2√3 + 2i = 4(cos 30° + i sin 30°)
  6. 2(cos 60° + i sin 60°) in rectangular form
    Answer
    = 1 + √3 i

Medium

  1. write -1 + i in polar form
    Answer
    = −1 + i = √2(cos 135° + i sin 135°)
  2. write -√3 + i in polar form
    Answer
    = −√3 + i = 2(cos 150° + i sin 150°)
  3. write -1 + i√3 in polar form
    Answer
    = −1 + √3 i = 2(cos 120° + i sin 120°)
  4. write 1 - i in polar form
    Answer
    = 1 − i = √2(cos 315° + i sin 315°)
  5. write √3 - i in polar form
    Answer
    = √3 − i = 2(cos 330° + i sin 330°)
  6. write -2 - 2i in polar form
    Answer
    = −2 − 2i = 2√2(cos 225° + i sin 225°)
  7. 4(cos 120° + i sin 120°) in rectangular form
    Answer
    = −2 + 2√3 i
  8. 6(cos 330° + i sin 330°) in rectangular form
    Answer
    = 3√3 − 3i

Harder

  1. (1 + i)^8
    Answer
    (1 + i)8 = 16
  2. (√3 + i)^6
    Answer
    (√3 + i)6 = −64
  3. (-1 + i)^4
    Answer
    (−1 + i)4 = −4
  4. (1 + i√3)^3
    Answer
    (1 + √3 i)3 = −8
  5. (1 - i)^6
    Answer
    (1 − i)6 = 8i
  6. (2 + 2i)^4
    Answer
    (2 + 2i)4 = −64
  7. (-√3 + i)^3
    Answer
    (−√3 + i)3 = 8i