Precalculus Polar Form and De Moivre’s Theorem Test with Point Values (PDF)

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

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This polar complex test 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.

This 20-point test covers polar and rectangular form, products, quotients, De Moivre powers and nth roots (N-CN.B.4 to N-CN.B.6). You have 45 minutes. Show your work.

Time and points

About this test

Grade: Precalculus
Chapter: Polar Form and De Moivre's Theorem
Standards: N-CN.B.4, N-CN.B.5, N-CN.B.6
Time: 45 minutes
Total: 20 points

Polar complex test: questions

Question 1 – Polar form [2 pts]

The number z.

Write \(2\sqrt3-2i\) in polar form with \(0^{\circ}\le\theta< 360^{\circ}\).

Question 2 – Rectangular form [2 pts]

Write \(8(\cos225^{\circ}+i\sin225^{\circ})\) in the form \(a+bi\).

Question 3 – Product [2 pts]

Multiply \(4\,\text{cis}\,40^{\circ}\) and \(2.5\,\text{cis}\,80^{\circ}\). Give polar and rectangular forms.

Question 4 – Quotient [2 pts]

Divide \(18\,\text{cis}\,280^{\circ}\) by \(6\,\text{cis}\,100^{\circ}\).

Question 5 – Power with De Moivre [3 pts]

The point z = -1 + i.

Let \(z=-1+i\). (a) Write \(z\) in polar form. (b) Use De Moivre’s theorem to find \(z^{6}\). (c) Write \(z^{6}\) in rectangular form.

Question 6 – Large power [2 pts]

Evaluate \((1-i)^{10}\).

Question 7 – Cube roots [2 pts]

Find the three cube roots of \(-27\) in polar form.

Question 8 – Fourth roots [3 pts]

Four equally spaced points.

(a) Write \(16\) in polar form. (b) Find the four fourth roots of \(16\). (c) Explain why they form a square.

Question 9 – Rotation [2 pts]

Rotate \(-2+5i\) by \(90^{\circ}\) counterclockwise by multiplying by \(i\).

Key facts behind this polar complex test

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 test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All Precalculus chapters: Precalculus math.

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