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4. Polar Form of a Complex Number

by M. Bourne

We can think of complex numbers as vectors, as in the above example.

We have met this idea before, in Polar Coordinates, part of the analytical geometry section.

From Pythagoras, we have: r² = x² + y² and basic trigonometry gives us:

x = r cos θy = r sin θ

Multiplying the last expression throughout by j gives us:

yj = jr sin θ

So we can write the polar form of a complex number as:

x + yj = r(cos θ + j sin θ)

r is the absolute value (or modulus) of the complex number

θ is the argument of the complex number.

There are two other ways of writing the polar form of a complex number:

r cis θ [means r (cos θ + j sin θ)]

r ∠ θ [means once again, r (cos θ + j sin θ)]

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NOTE: When writing a complex number in polar form, the angle θ can be in DEGREES or RADIANS.

Example 1

Find the polar form and represent graphically the complex number 7 - 5j.

Answer:

We need to find r and θ.

To find θ, we first find the acute angle α (see Trigonometric Functions of Any Angle if you are rusty on this):

Now, 7 - 5j is in the fourth quadrant, so

θ = 360° - 35.54° = 324.46°

So, expressing 7 - 5j in polar form, we have:

7 - 5j = 8.6 (cos 324.5° +j sin 324.5°)

We could also write this answer as 7 - 5j = 8.6 cis 324.5°.

Also we could write: 7 - 5j = 8.6 ∠ 324.5°

The graph is as follows:

Now let's see how it looks in Livemath.

LIVEMath

 

Example 2:

Express 3(cos 232°+ j sin 232°) in rectangular form.

We simply multiply out the expression:

3(cos 232° +j sin 232°)

= 3 cos 232° + j (3sin 232°)

= -1.85 - 2.36j

 

Exercises

1. Represent 1 + j √3 graphically and write it in polar form.

Answer:

We recognise this triangle as our 30-60 triangle from before.

θ = 60°, r = 2

So

1 + j√3 = 2 ∠ 60° = 2(cos60° + jsin60°)

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2. Represent √2 - j√2 graphically and write it in polar form.

Answer:

To find θ, we first find the acute angle α:

The complex number is in the 4th quadrant, so

θ = 360° - 45° = 315°

√2 - j√2 = 2∠ 315° = 2(cos315° + jsin315°)

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3. Represent graphically and give the rectangular form of
6(cos180°+ jsin180°).

Answer:

6(cos180° + jsin180°) = -6

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4. Represent graphically and give the rectangular form of 7.32 ∠ -270°

Answer

7.32 ∠ -270°= 7.32j

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Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• 5. Exponential Form
• 6. Products and Quotients
• 7. Powers and Roots
• 8. AC Circuit Definitions
• 9. Impedance and Phase Angle
• 10. Reactance and Angular Velocity
• 11. AC Circuit Exercises
• 12. Parallel AC Circuits
• Vocabulary
• Fractals

• Comments, Questions?

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