# Polar Form

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The format for the polar form is
$$\bbox[10px,border:1px solid grey]{C = Z\angle \theta} \tag{1}$$
with the letter $Z$ chosen from the sequence $X$, $Y$, $Z$ where $Z$ indicates magnitude only and $\theta$ is always measured counter-clockwise (CCW) from the positive real axis, as shown in [Fig. 1].
Fig. 1: Defining the polar form.
Fig. 2: Demonstrating the effect of a negative sign on the polar form.
Angles measured in the clockwise direction from the positive real axis must have a negative sign associated with them. A negative sign in front of the polar form has the effect shown in [Fig. 2]. Note that it results in a complex number directly opposite the complex number with a positive sign.
$$\bbox[10px,border:1px solid grey]{-C = -Z\angle \theta = Z \angle \theta\pm 180} \tag{2}$$
Example 1: Sketch the following complex numbers in the complex plane:
a. $C = 5 \angle 30^\circ$
b. $C = 7 \angle -120^\circ$
c. $C = -4.2 \angle 60^\circ$
Solutions:
a. See Fig. 3.
b. See Fig. 4.
c. See Fig. 5.
Fig. 3: Example 1 (a)
Fig. 4: Example 1 (b)
Fig. 5: Example 1 (c)

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