Monday, June 8, 2009

Trigonometric Form and deMoivre's Theorem

A complex number can be expressed in three ways.

The relationship between the three can be summarized with a diagram:

Converting from one form to the other involves using the right triangle to relate the sides to the angle.
Sometimes complex numbers are easier to work with in trigonometric form. For example, to multiply, divide, or take a power, it is often easier to convert to trig form and apply:


The third property above can be used to take the nth root of a complex number.

For
For example:


5 comments:

Nicole S. said...

for the last example, how come you add 120 degrees to x2 and x3?

Kate Nowak said...

Because since there are 3 roots, they are separated by 360/3 = 120 degrees.

If you were taking a 4th root, you would add 360/4 = 90 to each one. (etc)

Kelsey said...

I don't understand how you go from r = square root(a^2+b^2) to theta=inverse tan(a/b)
is there a formula or a step that you didn't show?

Kate Nowak said...

I'm not going from one to the other...

The first is how to get the radius if you know the rectangular coordinates. It's just the pythagorean theorem.

The second is how to get the angle if you know the rectangular coordinates. a and b are opposite and adjacent, so tangent is required.

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