4. Polar Form of a Complex Number
by M. Bourne
We can think of complex numbers as vectors, as in our earlier example. [See more on Vectors in 2-Dimensions].
We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section.
Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis.
We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and θ (the angle made with the real axis):
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)]
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:
Need Graph Paper?
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.
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°)
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°)
3. Represent graphically and give the rectangular form of
6(cos180°+
jsin180°).
Answer:
6(cos180° + jsin180°) = -6
4. Represent graphically and give the rectangular form of 7.32 ∠ -270°
Answer
7.32 ∠ -270°= 7.32j
And the good news is...
Now that you know what it all means, you can use your calculator directly to convert from rectangular to polar forms and in the other direction, too.
Click here for complex number examples using calculator.
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