Question

I dont understand why we graph polar coordinates in the complext plane, if x = rcos(theta) and y =rsin(theta) why cant we just convert all polar vectors to x and y? In my circuits class we're comverting polar to complex voltage, but I don't see why we dont just call the vertical component the y component.

Answer 1

See polar coordinates are quite different from the usual (x, y) points on the Cartesian coordinate system. Polar coordinates bring together both angle measures and distances, all in one neat package. With the polar coordinate system, you can graph curves that resemble flowers and hearts and other elegant shapes. Which can not be easily plotted with the help of cartesian coordinates.

We are work on complex numbers and polar coordinates in the following ways:

• Interpreting graphs of basic polar coordinates

• Graphing polar equations such as cardioids and lemniscates

Some times if we needed we will convert in reverse direction also. That is from polar to cartesian as well. Basically this will reduce the complexity of the graph either in polar or in cartesian.

That's why we use the change of coordinates.

The polar form of a complex number expresses a number in terms of an angle and its distance from the origin r. Given a complex number in the rectangular form expressed as z=x+yi, we use the same conversion formulas as we do to write the number in trigonometric form:

x = rcos

y = rsin

Making a direct substitution, we have

z = x + yi = (rcos) + i(rsin) = r(cos + isin)

where r is the modulus and is the argument. We often use the abbreviation r(cis) to represent r(cos + isinθ).