Precalculus Complex Numbers in Polar Form Worksheet with Answers – PDF

Precalculus Complex Polar Worksheet: problems 1–5

1. Plot and find the modulus

The figure shows four complex numbers. Write each one in the form \(a+bi\) and find its modulus.

2. Conjugates and products

Find the conjugate of each number and the product of the number with its conjugate.

1. \(3-2i\)
2. \(5+i\)
3. \(-4i\)
4. \(-2-6i\)

3. Dividing complex numbers

Write each quotient in the form \(a+bi\).

1. \(\frac{2+3i}{1-i}\)
2. \(\frac{5}{2+i}\)
3. \(\frac{3-i}{1+2i}\)
4. \(\frac{4+2i}{-i}\)

4. Finding the argument

Find the argument, between \(0^{\circ}\) and \(360^{\circ}\), of each number shown.

1. \(a=1+i\)
2. \(b=-1+i\)
3. \(c=-\sqrt{3}-i\)
4. \(d=2-2\sqrt{3}\,i\)

5. Rectangular to polar form

Write each number in polar form \(r(\cos\theta+i\sin\theta)\) with \(0^{\circ}\le\theta < 360^{\circ}\).

1. \(3\sqrt{3}+3i\)
2. \(-2+2i\)
3. \(-5i\)
4. \(1-\sqrt{3}\,i\)

Precalculus Complex Polar Worksheet: problems 6–10

6. Polar to rectangular form

Write each number in the form \(a+bi\). Give exact values.

1. \(4(\cos 60^{\circ}+i\sin 60^{\circ})\)
2. \(10(\cos 180^{\circ}+i\sin 180^{\circ})\)
3. \(6(\cos 225^{\circ}+i\sin 225^{\circ})\)
4. \(2\sqrt{2}(\cos 315^{\circ}+i\sin 315^{\circ})\)

7. Radian measure

Write each number in the form \(a+bi\).

1. \(2(\cos \frac{\pi}{6}+i\sin \frac{\pi}{6})\)
2. \(5(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2})\)
3. \(8(\cos \frac{5\pi}{3}+i\sin \frac{5\pi}{3})\)
4. \(6(\cos \frac{3\pi}{4}+i\sin \frac{3\pi}{4})\)

8. Conjugate in polar form

Let \(z=6(\cos 50^{\circ}+i\sin 50^{\circ})\).

1. Write \(\overline{z}\) in polar form with an angle between \(0^{\circ}\) and \(360^{\circ}\).
2. Find \(z\overline{z}\) using polar form.

9. Multiplying in polar form

Find each product. Write the result in polar form, then in the form \(a+bi\).

1. \(2(\cos 25^{\circ}+i\sin 25^{\circ})\cdot7(\cos 35^{\circ}+i\sin 35^{\circ})\)
2. \(3(\cos 110^{\circ}+i\sin 110^{\circ})\cdot4(\cos 70^{\circ}+i\sin 70^{\circ})\)
3. \(\sqrt{2}(\cos 200^{\circ}+i\sin 200^{\circ})\cdot\sqrt{2}(\cos 160^{\circ}+i\sin 160^{\circ})\)

10. Dividing in polar form

Find each quotient. Write the result in polar form, then in the form \(a+bi\).

1. \(\frac{12(\cos 150^{\circ}+i\sin 150^{\circ})}{3(\cos 30^{\circ}+i\sin 30^{\circ})}\)
2. \(\frac{10(\cos 80^{\circ}+i\sin 80^{\circ})}{5(\cos 125^{\circ}+i\sin 125^{\circ})}\)
3. \(\frac{6(\cos \pi+i\sin \pi)}{2(\cos \frac{\pi}{2}+i\sin \frac{\pi}{2})}\)

Precalculus Complex Polar Worksheet: problems 11–15

11. Check a product two ways

Let \(z_{1}=1+i\) and \(z_{2}=\sqrt{3}+i\).

1. Find \(z_{1}z_{2}\) by multiplying in rectangular form.
2. Write \(z_{1}\) and \(z_{2}\) in polar form and find the product in polar form.
3. Confirm that the two results agree by comparing the real parts to the nearest hundredth.

12. Rotation and scaling

The point \(z=4+2i\) is shown. Multiply \(z\) by \(2i\). Describe the effect on the modulus and the argument, and plot the result.

13. Modulus of products and quotients

Let \(z_{1}=3+4i\) and \(z_{2}=5-12i\).

1. Find \(z_{1}z_{2}\) and its modulus.
2. Show that \(|z_{1}z_{2}|=|z_{1}|\,|z_{2}|\).
3. Find \(|\frac{z_{1}}{z_{2}}|\) without dividing the complex numbers.

14. De Moivre's theorem: basics

Use De Moivre’s theorem to simplify. Give each answer in the form \(a+bi\).

1. \((\cos 20^{\circ}+i\sin 20^{\circ})^{9}\)
2. \([2(\cos 15^{\circ}+i\sin 15^{\circ})]^{4}\)
3. \((\cos\frac{\pi}{8}+i\sin\frac{\pi}{8})^{4}\)

15. Powers of complex numbers

Evaluate each power by converting to polar form first.

1. \((1+i)^{8}\)
2. \((1-i)^{6}\)
3. \((-1+\sqrt{3}\,i)^{3}\)
4. \((\sqrt{3}+i)^{5}\)

Precalculus Complex Polar Worksheet: problems 16–20

16. Distance in the complex plane

Find the distance between each pair of complex numbers.

1. \(1+2i\) and \(5+5i\)
2. \(-3+4i\) and \(2-8i\)
3. \(7-i\) and \(-1+5i\)

17. Midpoints

Complete each part.

1. Find the midpoint of \(1+2i\) and \(5+5i\).
2. Find the midpoint of \(-4+6i\) and \(2-2i\).
3. One endpoint of a segment is \(3-i\) and its midpoint is \(1+4i\). Find the other endpoint.

18. Triangle in the complex plane

The vertices of a triangle are the complex numbers \(0\), \(6\) and \(3+4i\). Show that the triangle is isosceles, then find its perimeter and area.

19. Circle in the complex plane

The set of points \(z\) with \(|z-(2+i)|=3\) is a circle. State its center and radius. Then decide whether each point lies on the circle, inside it, or outside it: \(5+i\), \(-1+i\), \(2+4i\), \(4+3i\).

20. Robot vacuum rotation

A robot vacuum is at the point \(z=2+3i\) (distances in feet) on a floor plan whose origin is the charging dock. It rotates about the dock by \(90^{\circ}\) counterclockwise. Where does it end up? Then it rotates the original position by \(30^{\circ}\) counterclockwise instead. Find that position to the nearest hundredth.

Precalculus Complex Polar Worksheet: problems 21–22

21. AC circuit impedance

In an AC circuit, an impedance of \(Z=8+6i\) ohms has a voltage of \(V=120\) volts across it. Find the modulus of \(Z\), its phase angle to the nearest hundredth of a degree, and the current \(I=\frac{V}{Z}\) in the form \(a+bi\). Then find the magnitude of the current.

22. Powers: multi-step

Complete each part.

1. Evaluate \((\sqrt{3}-i)^{6}\).
2. Find the smallest positive integer \(n\) for which \((\cos 40^{\circ}+i\sin 40^{\circ})^{n}=1\).
3. Find the smallest positive integer \(n\) for which \((\sqrt{3}+i)^{n}\) is a purely imaginary number.

Which problems build the conversion skills?

Problem 1 reads four plotted numbers and finds each modulus. Division comes next, as problem 3 asks for \(\frac{2+3i}{1-i}\) and similar quotients in \(a+bi\) form. Problem 4 then searches for arguments between \(0^{\circ}\) and \(360^{\circ}\). Pay attention to the quadrant, since a raw calculator angle often needs \(180^{\circ}\) added. Problem 7 repeats the work with radians.

How do the product and power problems differ?

Problem 11 checks a product two ways. With \(z_{1}=1+i\) and \(z_{2}=\sqrt{3}+i\), the moduli are \(\sqrt{2}\) and 2, and the angles are \(45^{\circ}\) and \(30^{\circ}\). Both routes must agree on \(2\sqrt{2}\) and \(75^{\circ}\). Likewise, problem 13 shows that \(|z_{1}z_{2}|=|z_{1}||z_{2}|\) for \(3+4i\) and \(5-12i\), where the moduli multiply to 65. In problem 15, convert before you raise to a power such as \((-1+\sqrt{3}\,i)^{3}\).

What do the plane and circuit problems ask?

Problem 16 measures distances such as the gap between \(-3+4i\) and \(2-8i\). Problem 18 proves that a triangle with vertices \(0\), \(6\) and \(3+4i\) is isosceles. In problem 19, a circle has center \(2+i\) and radius 3. Finally, problem 21 gives an impedance of \(8+6i\) ohms with 120 volts, so you divide to find the current.

Start each polar problem with a quick sketch. That habit prevents most quadrant errors.

How should you review your results?

Sort the missed problems into three groups: angles, arithmetic and meaning. Redo the angle problems first, because they affect everything after them. Afterward, check the key, and then take the quiz to see whether the fixes hold.

See every chapter on the Precalculus math page.