http://www.clarku.edu/~djoyce/trig/
Polar coordinates will help us understand complex numbers geometrically. On the one hand, the usual rectangular coordinates x and y specify a complex number z = x + yi by giving the distance x right and the distance y up. On the other hand, polar coordinates specify the same point z by saying how far r away from the origin 0, and the angle for the line from the origin to the point. We’ve already called the distance r the absolute value |z| of z, and we saw how the Pythagorean theorem gave relation between it and x and y:
Next, we need to deal with the angle . We’ll follow the standard convention for specifying the angle
. This convention takes the positive x-axis (our real axis) to be at angle 0°, the
positive y-axis (our imaginary axis) at angle 90°, the negative x-axis angle 180°, and the negative
y-axis at angle 270°. Also, 360° can be added or subtracted from any angle and the direction is not changes. So,
0°, 360°, 720°, and –360° all refer to the positive x-axis. Similarly, 270° and –90°
both refer to the negative y-axis. A 45° angle runs along the line y = x, up to the right.
And so forth.
A point z can be specified by either pair, the pair of rectangular coordinates, x and y, or the pair of polar coordinates, r, which is |z|, and , which is arg (z). Since either pair determines the point, each pair should determine the other pair. There should be four equations, connecting them, and so there are. The Pythagorean identity was mentioned above, but the others require trigonometry. From the same triangle we used for the Pythagorean theorem, we find the following three relations:
Now, if we apply these relations to our complex number z = x + yi, then we get an alternate description for z
z | = | x + iy |
= | r cos + i r sin | |
= | r (cos + i sin ) | |
= | |z| (cos + i sin ) |
Note that the complex number cos + i sin has absolute value 1 since cos^{2} + sin^{2} equals 1 for any angle . Thus, every complex number z is the product of a real number |z| and a complex number cos + i sin .
We’re almost to the point where we can prove the last unproved statement of the previous section on multiplication, namely, that arg(zw) = arg(z) + arg(w). As above, we take arg(z) to be , and now let arg(w) be . Then,
We need to show that arg(zw) is + . In other words
If we use the addition formulas for cosine and sine at one crucial point, we’ll have it. Recall from trigonometry these addition formulas:
sin ( + ) = cos sin + sin cos .
Now we’re ready to show arguments add in the product zw.
zw | = | |z| (cos + i sin ) |w| (cos + i sin ) |
= | |zw| (cos + i sin ) (cos + i sin ) | |
= | |zw| ((cos cos – sin sin ) + i(cos sin + sin cos )) | |
= | |zw| (cos ( + ) + i sin ( + )) |
Thus, arg(zw) is + , as claimed.