Trigonometric Form of a Complex Number

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Trigonometric Form of a Complex Number
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Complex Numbers
Argand Diagram
Recall that a complex number has a real component
and an imaginary component. z a bi
Imaginary axis
Real axis
a
bi
z 3 2i
z 3 2i
The absolute value of a complex number is its
distance from the origin.
The names and letters are changing, but this sure
looks familiar.
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The Trig form of a Complex Number
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How is it Different?
In a rectangular system, you go left or right and
up or down.
In a trigonometric or polar system, you have a
direction to travel and a distance to travel in
that direction.
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Converting from Rectangular form to Trig form
Convert z 4 3i to trig form.
1. Find r
2. Find
3. Fill in the blanks
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Converting from Trig Form to Rectangular Form

• This ones easy.
• Evaluate the sin and cos.
• Distribute in r

Convert 4(cos 30 i sin 30) to rectangular form.
1. Evaluate the sin and cos
2. Distribute the 4.
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Multiplying Complex Numbers
To multiply complex numbers in rectangular form,
you would FOIL and convert i2 into 1.
To multiply complex numbers in trig form, you
simply multiply the rs and add the thetas.
The formulas are scarier than it really is.
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Example
Rectangular form
Trig form
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Dividing Complex Numbers
In rectangular form, you rationalize using the
complex conjugate.
In trig form, you just divide the rs and subtract
the theta.
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Example
Rectangular form
Trig form
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De Moivres Theorem

• If is a
  complex number
• And n is a positive integer
• Then

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Who was De Moivre?
A brilliant French mathematician who was
persecuted in France because of his religious
beliefs. De Moivre moved to England where he
tutored mathematics privately and became friends
with Sir Issac Newton.
De Moivre made a breakthrough in the field of
probability (writing the Doctrine of Chance), but
more importantly he moved trigonometry into the
field of analysis through complex numbers with De
Moivres theorem.
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But, can we prove DeMoivres Theorem?
Lets look at some Powers of z.
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Lets look at some more Powers of z.
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It appears that
Proof
Assume n1, then the statement is true. We can
continue in the previous manor up to some
arbitrary k Let n k, so that
Now find
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Eulers Formula
We can also use Eulers formula to prove
DeMoivres Theorem.
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So what is the use?
Find an identity for using Mr. De
Moivres fantastic theory
Remember the binomial expansion
Apply it
Cancel out the imaginery numbers
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Now try these
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Powers of Complex Numbers
This is horrible in rectangular form.
Its much nicer in trig form. You just raise the
r to the power and multiply theta by the exponent.
The best way to expand one of these is using
Pascals triangle and binomial expansion. Youd
need to use an i-chart to simplify.
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Roots of Complex Numbers

• There will be as many answers as the index of the
  root you are looking for
• Square root 2 answers
• Cube root 3 answers, etc.
• Answers will be spaced symmetrically around the
  circle
• You divide a full circle by the number of answers
  to find out how far apart they are

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The formula
Using DeMoivres Theorem we get
k starts at 0 and goes up to n-1 This is easier
than it looks.
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General Process

1. Problem must be in trig form
2. Take the nth root of r. All answers have the
   same value for r.
3. Divide theta by n to find the first angle.
4. Divide a full circle by n to find out how much
   you add to theta to get to each subsequent answer.

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Example
1. Find the 4th root of 81
2. Divide theta by 4 to get the first angle.
3. Divide a full circle (360) by 4 to find out
how far apart the answers are.

• List the 4 answers.
• The only thing that changes is the angle.
• The number of answers equals the number of roots.