Trigonometric Form of Complex Numbers - Concept
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.
Please enter your name.
Are you sure you want to delete this comment?
- Converting Complex Numbers From Trigonometric Form to Rectangular 3,526 views
- Converting Complex Numbers From Rectangular Form to Trigonometric 14,267 views
- Multiplying Complex Numbers 7,883 views
- Dividing Complex Numbers 7,164 views
- DeMoivre's Theorem 15,045 views
- The Euler Formula 10,184 views
- Finding the Roots of a Complex Number 28,395 views
- More Roots of Complex Numbers 10,814 views
- Introduction to Polar Coordinates 33,323 views
- Converting from Rectangular Coordinates to Polar 40,429 views
- Converting from Polar Coordinates to Rectangular 27,085 views
- The Distance Formula in Polar Coordinates 16,206 views
- Lines in Polar Coordinates 12,464 views
- Symmetry of Polar Graphs 20,024 views
- Graphing Polar Equations 28,117 views
- Families of Polar Curves: Circles, Cardiods, and Limacon 11,487 views
- Families of Polar Curves: Roses 12,517 views
- Families of Polar Curves: Conic Sections 7,664 views
- The Complex Plane 10,957 views