Trigonometric Form of Complex Numbers

The trigonometric form of a complex number (also called polar form) writes a + bi as r(\cos \theta + i \sin \theta). Here r = \sqrt{a^2 + b^2} is the modulus, the distance from 0, and \theta is the argument, the angle from the positive real axis. This form makes multiplying and dividing complex numbers easy.

Before this lesson: You should know Operations with Complex Numbers, Polar Coordinates and Reference Angles.

What Trigonometric Form Means

Plot a + bi in the complex plane: go a units along the real (horizontal) axis and b units along the imaginary (vertical) axis. This is just like plotting the point (a, b).

Now describe that point with polar coordinates instead. Its distance from 0 is r, and its direction is \theta. Since a = r \cos \theta and b = r \sin \theta, the number is r \cos \theta + (r \sin \theta) i, which factors to r(\cos \theta + i \sin \theta).

Formulas

z = a + bi = r(\cos \theta + i \sin \theta)

• Modulus: r = \lvert z \rvert = \sqrt{a^2 + b^2}
• Argument: \tan \theta = \frac{b}{a}, with \theta in the same quadrant as (a, b)
• Some books write r \operatorname{cis} \theta as a short way to write r(\cos \theta + i \sin \theta).

Multiplying and dividing

If z_1 = r_1(\cos \theta_1 + i \sin \theta_1) and z_2 = r_2(\cos \theta_2 + i \sin \theta_2), then:

z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]

\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]

In words: to multiply, multiply the moduli and add the angles. To divide, divide the moduli and subtract the angles. This comes from the sum and difference formulas.

Worked Examples

Example 1: Write a number in trigonometric form

Write 1 + i\sqrt{3} in trigonometric form, with 0 \le \theta \lt 2\pi.

Identify a = 1 and b = \sqrt{3}

r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2

\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}

The point (1, \sqrt{3}) is in Quadrant I, so \theta = \frac{\pi}{3}

Answer: 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}), as in the picture above.

Example 2: A number in Quadrant III

Write -4 - 4i in trigonometric form.

r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32}

\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}

Find the tangent: \tan \theta = \frac{-4}{-4} = 1, so the reference angle is \frac{\pi}{4}

\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}

Answer: 4\sqrt{2}(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}). A calculator gives \tan^{-1}(1) = \frac{\pi}{4}, which is in the wrong quadrant.

Example 3: Back to standard form

Write 6(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}) in the form a + bi.

Use the unit circle: \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} and \sin \frac{5\pi}{6} = \frac{1}{2}

6(-\frac{\sqrt{3}}{2} + \frac{1}{2} i)

-\frac{6\sqrt{3}}{2} + \frac{6}{2} i

-3\sqrt{3} + 3i

Answer: -3\sqrt{3} + 3i, which is about -5.20 + 3i

Example 4: Multiply in trigonometric form

Multiply z_1 = 3(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) and z_2 = 2(\cos \frac{\pi}{12} + i \sin \frac{\pi}{12}). Write the answer in both forms.

3 \cdot 2 = 6

\frac{\pi}{4} + \frac{\pi}{12} = \frac{3\pi}{12} + \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}

z_1 z_2 = 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})

6(\frac{1}{2} + \frac{\sqrt{3}}{2} i)

3 + 3\sqrt{3} i

Answer: 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) = 3 + 3\sqrt{3} i

Example 5: Word problem with an AC circuit

In an AC circuit, current I, voltage V and impedance Z are complex numbers with I = \frac{V}{Z}. A circuit has V = 120(\cos 0^\circ + i \sin 0^\circ) volts and Z = 40(\cos 30^\circ + i \sin 30^\circ) ohms. Find the current in both forms.

\frac{120}{40} = 3

0^\circ - 30^\circ = -30^\circ

I = 3(\cos(-30^\circ) + i \sin(-30^\circ))

I = 3(\frac{\sqrt{3}}{2} - \frac{1}{2} i)

I = \frac{3\sqrt{3}}{2} - \frac{3}{2} i \approx 2.60 - 1.5i

Answer: I = 3(\cos(-30^\circ) + i \sin(-30^\circ)) \approx 2.60 - 1.5i amps. The current has size 3 amps and lags the voltage by 30^\circ.

Common Mistakes

Related Methods

Powers and roots

Multiplying a number by itself many times means multiplying the modulus many times and adding the angle many times. That shortcut for powers and roots is De Moivre’s theorem.

Standard form for adding

Trigonometric form is great for multiplying and dividing, but not for adding. To add or subtract, convert to a + bi first and use operations with complex numbers.

Non-special angles

If \frac{b}{a} is not a unit circle value, use the inverse tangent (see inverse trig functions) for the reference angle, then adjust for the quadrant.

Practice Problems

Use 0 \le \theta \lt 2\pi unless the problem uses degrees.

1. Write 3 + 3i in trigonometric form.
2. Write -2\sqrt{3} + 2i in trigonometric form.
3. Write -5i in trigonometric form.
4. Write 7 in trigonometric form.
5. Write 4(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) in the form a + bi.
6. Write 3 - 4i in trigonometric form. Round \theta to the nearest hundredth of a radian.
7. Multiply 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) by 5(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}). Give the answer as a + bi.
8. Divide 12(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) by 3(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}). Give the answer as a + bi.
9. Multiply 4(\cos 70^\circ + i \sin 70^\circ) by 1.5(\cos 50^\circ + i \sin 50^\circ). Give the answer as a + bi.
10. A circuit has V = 240(\cos 45^\circ + i \sin 45^\circ) volts and Z = 60(\cos 15^\circ + i \sin 15^\circ) ohms. Find I = \frac{V}{Z} in both forms.

r = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}

r = \sqrt{(-2\sqrt{3})^2 + 2^2} = \sqrt{12 + 4} = \sqrt{16} = 4

\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}

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

-2 + 2\sqrt{3} i

r = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

\tan^{-1}(\frac{4}{3}) \approx 0.9273

\theta \approx 2\pi - 0.9273 \approx 5.36

2 \cdot 5 = 10

\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}

10(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 10(0 + 1i)

\frac{12}{3} = 4

\frac{3\pi}{4} - \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2}

4(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 4(0 + 1i)

4 \cdot 1.5 = 6

70^\circ + 50^\circ = 120^\circ

6(\cos 120^\circ + i \sin 120^\circ) = 6(-\frac{1}{2} + \frac{\sqrt{3}}{2} i)

-3 + 3\sqrt{3} i

\frac{240}{60} = 4

45^\circ - 15^\circ = 30^\circ

I = 4(\cos 30^\circ + i \sin 30^\circ) = 4(\frac{\sqrt{3}}{2} + \frac{1}{2} i)

I = 2\sqrt{3} + 2i

FAQ

What is the trigonometric form of a complex number?

It is r(\cos \theta + i \sin \theta), where r is the distance from 0 in the complex plane and \theta is the angle from the positive real axis.

How do you find the modulus and argument?

The modulus is r = \sqrt{a^2 + b^2}. The argument is the angle with \tan \theta = \frac{b}{a}, placed in the same quadrant as the point (a, b).

How do you multiply complex numbers in trigonometric form?

Multiply the moduli and add the arguments. To divide, divide the moduli and subtract the arguments.

Is polar form the same as trigonometric form?

Yes. Both names mean r(\cos \theta + i \sin \theta), because the number is described by polar coordinates (r, \theta).

Summary

• a + bi = r(\cos \theta + i \sin \theta), with r = \sqrt{a^2 + b^2}.
• Choose \theta from \tan \theta = \frac{b}{a} in the correct quadrant.
• To go back, find r \cos \theta and r \sin \theta.
• Multiply: multiply the moduli, add the angles.
• Divide: divide the moduli, subtract the angles.

Previous lesson: Graphing Polar Equations

Next lesson: De Moivre’s Theorem

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