Trigonometric Form of a Complex Number

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Trigonometric Form of a Complex Number

Complex Numbers Argand Diagram
Recall that a complex number has a real component and an imaginary component. z = a + bi Imaginary axis a Real axis 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.

The Trig form of a Complex Number

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.

Converting from Rectangular form to Trig form
Convert z = 4 + 3i to trig form. 1. Find r 2. Find 3. Fill in the blanks

Converting from Trig Form to Rectangular Form
This one’s 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.

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.

Example Rectangular form Trig form

Dividing Complex Numbers
In rectangular form, you rationalize using the complex conjugate. In trig form, you just divide the rs and subtract the theta.

Example Rectangular form Trig form

De Moivre’s Theorem If is a complex number And n is a positive integer
Then

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 Moivre’s theorem.

But, can we prove DeMoivre’s Theorem?
Let’s look at some Powers of z.

Let’s look at some more Powers of z.

It appears that: Proof: Assume n=1, then the statement is true.
We can continue in the previous manor up to some arbitrary k Let n = k, so that: Now find

Euler’s Formula We can also use Euler’s formula to prove DeMoivre’s Theorem.

So what is the use? Find an identity for using Mr. De Moivre’s fantastic theory Remember the binomial expansion: Apply it: Cancel out the imaginery numbers:

Now try these:

Powers of Complex Numbers
This is horrible in rectangular form. It’s 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 Pascal’s triangle and binomial expansion. You’d need to use an i-chart to simplify.

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

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

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

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.

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