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Trigonometric Form of Complex Numbers - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

A convenient form for numbers in the complex plane, other than rectangular form, is the trigonometric form of complex numbers. The trigonometric form of complex numbers uses the modulus and an angle to describe a complex number's location. It is important to be able to convert from rectangular to trigonometric form of complex numbers and from trigonometric to rectangular form.

We're talking about the complex plane and let me remind you, the way we visualize complex numbers, is by plotting in the plane, we graph the number z equals a+bi by plotting the point a coma b and so that means that this point is a units to the right and b units up from the origin. In the copmplex plane this is the real axis and this is the imaginary axis.
I want to define the term the absolute value or modulus of z. This complex number is defined as the absolute value of z is the square root of a squared plus b squared. Now this comes right from the Pythagorean Theorem. This quantity is exactly the length of this red segment. It's the distance of z from the origin and that's exactly what absolute value is for real numbers, the distance from the origin. So for complex numbers it means the same thing. Sometimes the term modulus is used for absolute value, but it means absolute value of z.
Now in addition to this kind of form for complex numbers, there is another form that's kind of reminiscent of polar coordinates. Now if we call this length r and we measure an angle form the real axis theta, we can actually put z in terms of r and theta in the following way. First of all let's observe that r is exactly equal to the modulus of z. r equals the square root of a squared plus b squared and secondly, let's observe that that cosine of theta is a over r. So a equals r cosine theta and sine theta is b over r. So b equals r sine theta and so we can rewrite this form which is actually called rectangular form in another way, z can be written. Since a is our cosine theta, I'll put r cosine theta in place of a plus b is our sine theta. I put r sine theta times i. Now the convention is to pull the r out and put parentheses. Cosine theta plus and you put the i in front, you don't want it to look like you're taking the sine of theta times i. So i sine theta. This form is called "Trigonometric form." sometimes called "polar form" and it's really useful as you'll see in a lesson or two. But we'll be using trigonometric form a lot and what we're going to do in the near future is learn how to convert back and forth between trigonometric form and rectangular form.

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