Precalculus Polar Form and De Moivre’s Theorem Worksheet (PDF)

Precalculus math worksheet: Polar Form and De Moivre's Theorem – Maths PDF

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This polar complex worksheet for Precalculus covers polar form and de moivre's theorem (Common Core N-CN.B.4, N-CN.B.5, N-CN.B.6), with a free PDF to print.

These 20 problems practice polar form and De Moivre’s theorem (Common Core N-CN.B.4, N-CN.B.5 and N-CN.B.6). You will convert between forms, multiply and divide in polar form, find powers and find nth roots of complex numbers.

Precalculus · Common Core State Standards: N-CN.B.4, N-CN.B.5, N-CN.B.6

Polar complex worksheet: exercises 1 to 10

Exercise 1 – Modulus and argument

The point z in the plane.

Find the modulus and argument of \(z=1+\sqrt3\,i\) and write \(z\) in polar form.

Exercise 2 – Quadrant II

Write \(-3+3i\) in polar form.

Exercise 3 – On an axis

Write \(-4i\) in polar form.

Exercise 4 – Polar to rectangular

Write \(6(\cos120^{\circ}+i\sin120^{\circ})\) in the form \(a+bi\).

Exercise 5 – Polar to rectangular in radians

Write \(4\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)\) in the form \(a+bi\).

Exercise 6 – Quadrant IV

Write \(5-5i\) in polar form.

Exercise 7 – Multiply in polar form

Two complex numbers and their product.

Multiply \(2(\cos30^{\circ}+i\sin30^{\circ})\) and \(3(\cos45^{\circ}+i\sin45^{\circ})\). Give the answer in polar and rectangular form (three decimals).

Exercise 8 – Divide in polar form

Divide \(12\,\text{cis}\,200^{\circ}\) by \(4\,\text{cis}\,50^{\circ}\).

Exercise 9 – Check with rectangular multiplication

Multiply \((1+i)(\sqrt3+i)\) in polar form and check in rectangular form.

Exercise 10 – De Moivre with a unit modulus

Evaluate \((\cos20^{\circ}+i\sin20^{\circ})^{9}\).

Exercises 11 and up

Exercise 11 – Power of 1 + i

Evaluate \((1+i)^{8}\).

Exercise 12 – Power with a negative angle

Evaluate \((\sqrt3-i)^{6}\).

Exercise 13 – Power in rectangular form

Evaluate \(\left[2(\cos15^{\circ}+i\sin15^{\circ})\right]^{4}\) in the form \(a+bi\).

Exercise 14 – Cube roots of 8

Three points on a circle.

Find the three cube roots of \(8\) in polar and rectangular form.

Exercise 15 – Sixth roots of unity

A hexagon on the unit circle.

List the sixth roots of unity as \(\text{cis}\,\theta\).

Exercise 16 – Fourth roots of -16

Find the four fourth roots of \(-16\) in rectangular form.

Exercise 17 – Solve z cubed = 27i

Solve \(z^{3}=27i\).

Exercise 18 – AC circuit

An impedance is \(Z=3+4i\) ohms and the current is \(I=10\) amps (modulus 10, angle \(0^{\circ}\)). Write \(Z\) in polar form, then find the voltage \(V=IZ\) in polar form.

Exercise 19 – Rotate a point

A rotation of 60 degrees.

Multiplying by \(\text{cis}\,60^{\circ}\) rotates a point \(60^{\circ}\) counterclockwise about the origin. Rotate \(4+3i\) by \(60^{\circ}\) and give the result to two decimals.

Exercise 20 – Cube root of unity

Let \(\omega=-\frac12+\frac{\sqrt3}{2}i\). Show that \(\omega^{3}=1\) and \(1+\omega+\omega^{2}=0\).

Key facts behind this polar complex worksheet

Key facts

z = a + bi = r(cos theta + i sin theta) with r = sqrt(a^2 + b^2) and tan theta = b/a.
Product: multiply moduli, add arguments. Quotient: divide moduli, subtract arguments.
De Moivre: [r cis theta]^n = r^n cis(n theta).
nth roots: (r)^(1/n) cis((theta + 360 k)/n), k = 0, …, n – 1.
The n roots form a regular n-gon on a circle of radius r^(1/n).

How to use this polar complex worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All Precalculus chapters: Precalculus math.

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