De Moivre's theorem worksheet

Raising a complex number to a power is easy in polar form: the modulus is raised to the power and the angle is multiplied by it. Doing the same thing by repeated multiplication in standard form is where the afternoon goes.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name De Moivre's Theorem Date Period Simplify using De Moivre's theorem. Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{5}$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{5}$ Find ${[(\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})]}^{2}$ Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ Find ${[(\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ Find ${[(\cos 0 + i \sin 0)]}^{4}$ Find ${[2 (\cos 0 + i \sin 0)]}^{5}$ Find ${[2 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})]}^{3}$ Find ${[(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})]}^{4}$ Find ${[2 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})]}^{3}$ Find ${[(\cos \frac{π}{2} + i \sin \frac{π}{2})]}^{5}$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ Find ${[2 (\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{3}$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{3}$ Find ${[(\cos \frac{π}{4} + i \sin \frac{π}{4})]}^{4}$ Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{2}$ Find ${[(\cos \frac{π}{6} + i \sin \frac{π}{6})]}^{3}$ Find ${[(\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})]}^{5}$ Find ${[(\cos 0 + i \sin 0)]}^{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

De Moivre's Theorem

Date Period

Simplify using De Moivre's theorem.

Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{5}$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{5}$
Find ${[(\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})]}^{2}$
Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$
Find ${[(\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$
Find ${[(\cos 0 + i \sin 0)]}^{4}$
Find ${[2 (\cos 0 + i \sin 0)]}^{5}$
Find ${[2 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})]}^{3}$
Find ${[(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})]}^{4}$
Find ${[2 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})]}^{3}$
Find ${[(\cos \frac{π}{2} + i \sin \frac{π}{2})]}^{5}$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$
Find ${[2 (\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$
Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{3}$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{3}$
Find ${[(\cos \frac{π}{4} + i \sin \frac{π}{4})]}^{4}$
Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{2}$
Find ${[(\cos \frac{π}{6} + i \sin \frac{π}{6})]}^{3}$
Find ${[(\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})]}^{5}$
Find ${[(\cos 0 + i \sin 0)]}^{2}$



Sigma Prep · sigmaprep.io/worksheets Name De Moivre's TheoremAnswer keyVersion 1 Date Period Simplify using De Moivre's theorem. Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{5}$ $32 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{5}$ $32 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})$ Find ${[(\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})]}^{2}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$ Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ $(\cos \frac{5 π}{3} + i \sin \frac{5 π}{3})$ Find ${[(\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ $(\cos π + i \sin π)$ Find ${[(\cos 0 + i \sin 0)]}^{4}$ $(\cos 0 + i \sin 0)$ Find ${[2 (\cos 0 + i \sin 0)]}^{5}$ $32 (\cos 0 + i \sin 0)$ Find ${[2 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})]}^{3}$ $8 (\cos \frac{π}{2} + i \sin \frac{π}{2})$ Find ${[(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})]}^{4}$ $(\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})$ Find ${[2 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})]}^{3}$ $8 (\cos 0 + i \sin 0)$ Find ${[(\cos \frac{π}{2} + i \sin \frac{π}{2})]}^{5}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ $16 (\cos π + i \sin π)$ Find ${[2 (\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ $32 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3})$ Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{3}$ $(\cos π + i \sin π)$ Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{3}$ $8 (\cos \frac{π}{4} + i \sin \frac{π}{4})$ Find ${[(\cos \frac{π}{4} + i \sin \frac{π}{4})]}^{4}$ $(\cos π + i \sin π)$ Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{2}$ $4 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})$ Find ${[(\cos \frac{π}{6} + i \sin \frac{π}{6})]}^{3}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$ Find ${[(\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})]}^{5}$ $(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})$ Find ${[(\cos 0 + i \sin 0)]}^{2}$ $(\cos 0 + i \sin 0)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

De Moivre's TheoremAnswer keyVersion 1

Date Period

Simplify using De Moivre's theorem.

Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{5}$ $32 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{5}$ $32 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})$
Find ${[(\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})]}^{2}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$
Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ $(\cos \frac{5 π}{3} + i \sin \frac{5 π}{3})$
Find ${[(\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ $(\cos π + i \sin π)$
Find ${[(\cos 0 + i \sin 0)]}^{4}$ $(\cos 0 + i \sin 0)$
Find ${[2 (\cos 0 + i \sin 0)]}^{5}$ $32 (\cos 0 + i \sin 0)$
Find ${[2 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})]}^{3}$ $8 (\cos \frac{π}{2} + i \sin \frac{π}{2})$
Find ${[(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})]}^{4}$ $(\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})$
Find ${[2 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})]}^{3}$ $8 (\cos 0 + i \sin 0)$
Find ${[(\cos \frac{π}{2} + i \sin \frac{π}{2})]}^{5}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{4}$ $16 (\cos π + i \sin π)$
Find ${[2 (\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{5}$ $32 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3})$
Find ${[(\cos \frac{π}{3} + i \sin \frac{π}{3})]}^{3}$ $(\cos π + i \sin π)$
Find ${[2 (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})]}^{3}$ $8 (\cos \frac{π}{4} + i \sin \frac{π}{4})$
Find ${[(\cos \frac{π}{4} + i \sin \frac{π}{4})]}^{4}$ $(\cos π + i \sin π)$
Find ${[2 (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})]}^{2}$ $4 (\cos \frac{3 π}{2} + i \sin \frac{3 π}{2})$
Find ${[(\cos \frac{π}{6} + i \sin \frac{π}{6})]}^{3}$ $(\cos \frac{π}{2} + i \sin \frac{π}{2})$
Find ${[(\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})]}^{5}$ $(\cos \frac{7 π}{6} + i \sin \frac{7 π}{6})$
Find ${[(\cos 0 + i \sin 0)]}^{2}$ $(\cos 0 + i \sin 0)$

How to do these

${(2 (\cos 3 0^{\circ} + i \sin 3 0^{\circ}))}^{3}$

The modulus 2 is cubed, giving 8.
The angle 30 is multiplied by 3, giving 90.

The answer is $8 (\cos 9 0^{\circ} + i \sin 9 0^{\circ})$ .

Where students go wrong

Multiplying the modulus by the power instead of raising it. A modulus of 2 cubed is 8, not 6. The angle is the one that multiplies.

Also called de moivre, powers of complex numbers or complex numbers in polar form.

Questions about these worksheets

Yes. Every de moivre's theorem 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 de moivre's theorem 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.

Multiplying the modulus by the power instead of raising it. A modulus of 2 cubed is 8, not 6. The angle is the one that multiplies.

De Moivre's theorem worksheet

Select your difficulty

De Moivre's Theorem

De Moivre's TheoremAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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