Worksheets · Precalculus

Complex numbers in polar form worksheet

A complex number is a point, so it has a distance and an angle like any other. Written that way, multiplying means multiplying the distances and adding the angles, which is what makes powers easy.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Complex Numbers and Polar Form

Date Period

Write each answer in the form asked for. Leave answers exact.

  1. Write in trigonometric form

    Write in trigonometric form with 0∘≤θ<360∘: −1+3i
  2. Write in trigonometric form with 0∘≤θ<360∘: 2+23i
  3. Write in trigonometric form with 0∘≤θ<360∘: 53+5i
  4. Write in trigonometric form with 0∘≤θ<360∘: −43+4i
  5. Write in trigonometric form with 0∘≤θ<360∘: −42+42i

  1. Write in standard form

    Write in standard form:
    10 ⁣(cos⁡210∘+isin⁡210∘)
  2. Write in standard form:
    6 ⁣(cos⁡45∘+isin⁡45∘)
  3. Write in standard form:
    8 ⁣(cos⁡120∘+isin⁡120∘)
  4. Write in standard form:
    4 ⁣(cos⁡300∘+isin⁡300∘)
  5. Write in standard form:
    10 ⁣(cos⁡315∘+isin⁡315∘)

  1. Use De Moivre's theorem

    Use De Moivre’s theorem to find (cos⁡330∘+isin⁡330∘)4. Give it in trigonometric form, or in standard form if it is real or imaginary.
  2. Let z=3 ⁣(cos⁡45∘+isin⁡45∘). Find z2. Give it in trigonometric form, or in standard form if it is real or imaginary.
  3. Find (cos⁡150∘+isin⁡150∘)3. Give it in trigonometric form, or in standard form if it is real or imaginary.
  4. Use De Moivre’s theorem to find [3 ⁣(cos⁡120∘+isin⁡120∘)]2. Give it in trigonometric form, or in standard form if it is real or imaginary.
  5. Find [2 ⁣(cos⁡210∘+isin⁡210∘)]4. Give it in trigonometric form, or in standard form if it is real or imaginary.

  1. Find roots in standard form

    Find all the cube roots of 27 in standard form
  2. Find all the cube roots of −27i in standard form
  3. Find all the square roots of −9 in standard form
  4. Find all the square roots of 50−503i in standard form
  5. Find all the cube roots of −i in standard form

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Complex Numbers and Polar FormAnswer keyVersion 1

Date Period

Write each answer in the form asked for. Leave answers exact.

  1. Write in trigonometric form

    Write in trigonometric form with 0∘≤θ<360∘: −1+3i2 ⁣(cos⁡120∘+isin⁡120∘)
  2. Write in trigonometric form with 0∘≤θ<360∘: 2+23i4 ⁣(cos⁡60∘+isin⁡60∘)
  3. Write in trigonometric form with 0∘≤θ<360∘: 53+5i10 ⁣(cos⁡30∘+isin⁡30∘)
  4. Write in trigonometric form with 0∘≤θ<360∘: −43+4i8 ⁣(cos⁡150∘+isin⁡150∘)
  5. Write in trigonometric form with 0∘≤θ<360∘: −42+42i8 ⁣(cos⁡135∘+isin⁡135∘)

  1. Write in standard form

    Write in standard form:
    10 ⁣(cos⁡210∘+isin⁡210∘)    −53−5i
  2. Write in standard form:
    6 ⁣(cos⁡45∘+isin⁡45∘)    32+32i
  3. Write in standard form:
    8 ⁣(cos⁡120∘+isin⁡120∘)    −4+43i
  4. Write in standard form:
    4 ⁣(cos⁡300∘+isin⁡300∘)    2−23i
  5. Write in standard form:
    10 ⁣(cos⁡315∘+isin⁡315∘)    52−52i

  1. Use De Moivre's theorem

    Use De Moivre’s theorem to find (cos⁡330∘+isin⁡330∘)4. Give it in trigonometric form, or in standard form if it is real or imaginary.cos⁡240∘+isin⁡240∘
  2. Let z=3 ⁣(cos⁡45∘+isin⁡45∘). Find z2. Give it in trigonometric form, or in standard form if it is real or imaginary.9i
  3. Find (cos⁡150∘+isin⁡150∘)3. Give it in trigonometric form, or in standard form if it is real or imaginary.i
  4. Use De Moivre’s theorem to find [3 ⁣(cos⁡120∘+isin⁡120∘)]2. Give it in trigonometric form, or in standard form if it is real or imaginary.9 ⁣(cos⁡240∘+isin⁡240∘)
  5. Find [2 ⁣(cos⁡210∘+isin⁡210∘)]4. Give it in trigonometric form, or in standard form if it is real or imaginary.16 ⁣(cos⁡120∘+isin⁡120∘)

  1. Find roots in standard form

    Find all the cube roots of 27 in standard form3, −32+332i, −32−332i
  2. Find all the cube roots of −27i in standard form3i, −332−32i, 332−32i
  3. Find all the square roots of −9 in standard form3i, −3i
  4. Find all the square roots of 50−503i in standard form−53+5i, 53−5i
  5. Find all the cube roots of −i in standard formi, −32−12i, 32−12i

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Write in trigonometric form: −23+2i

  1. The modulus is the square root of 12 plus 4, which is 4.
  2. The point is up and to the left, so the angle is 150 degrees.
  3. Write it as 4 times cos plus i sin of that angle.

The answer is 4 ⁣(cos⁡150∘+isin⁡150∘).

Where students go wrong

Taking the argument straight from the calculator. The modulus is always positive, so the angle is what carries the quadrant, and arctangent will not put you in the second or third one on its own.

Also called trigonometric form, modulus and argument or cis form.

What you can put on this worksheet

Standard form to trigonometric form
Write in trigonometric form with 0∘≤θ<360∘: −3+33i → 6 ⁣(cos⁡120∘+isin⁡120∘)
Trigonometric form to standard form
Write in standard form:
6 ⁣(cos⁡210∘+isin⁡210∘) → −33−3i
Multiplying and dividing
Multiply:
4 ⁣(cos⁡120∘+isin⁡120∘)⋅2 ⁣(cos⁡120∘+isin⁡120∘) → 8 ⁣(cos⁡240∘+isin⁡240∘)
De Moivre's theorem
Find [3 ⁣(cos⁡210∘+isin⁡210∘)]3. Give it in trigonometric form, or in standard form if it is real or imaginary. → −27i
Powers in standard form
Find (−2)3 → −8
Exponential form (Euler's formula)
Write in standard form:
6ei2π3 → −3+33i
The n roots of a complex number
Find the cube roots of 8 ⁣(cos⁡180∘+isin⁡180∘) → 2 ⁣(cos⁡60∘+isin⁡60∘), 2 ⁣(cos⁡180∘+isin⁡180∘), 2 ⁣(cos⁡300∘+isin⁡300∘)
Roots in standard form
Find all the cube roots of −8 in standard form → 1+3i, −2, 1−3i
Powers of a + bi, not special
(3+i)2 → 8+6i
ln of negatives and i, and i to an imaginary power
Evaluate (principal value): i3i → e−3π/2
e^z equations over the complex numbers
Find every complex solution: 2−ez=12. → z=ln⁡10+(2k+1)πi, k an integer
Plot on a polar grid
Plot z=2 ⁣(cos⁡7π6+isin⁡7π6) on a polar grid. — a blank polar grid → A: 2 out along 7π6

Questions about these worksheets

Yes. Every complex numbers in polar form sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the complex numbers in polar form questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 12: standard form to trigonometric form, trigonometric form to standard form, multiplying and dividing, de moivre's theorem, powers in standard form, exponential form (euler's formula), the n roots of a complex number, roots in standard form, powers of a + bi, not special, ln of negatives and i, and i to an imaginary power, e^z equations over the complex numbers and plot on a polar grid. Tick as many as you want and set how many of each, or let it spread them evenly.

Taking the argument straight from the calculator. The modulus is always positive, so the angle is what carries the quadrant, and arctangent will not put you in the second or third one on its own.