CN - Polar Form Lesson

Polar Form

So far, we've dealt with rectangular representations of complex numbers z = a + bi. However, we can also represent complex numbers in polar form.

Consider the complex number z = 3 + 2i, we know that to graph that, we move 3 units to the right and 2 units up. We can also write that number in polar form using r (the modulus of z) and θ, the direction of the point (starting at the positive x-axis).

image of an angle/triangle on graph

Let's consider a general case:

image of triangle abr plotted on graph with point (a, b)

By the Pythagorean theorem, we know that: equation image indicator LaTeX: r^2=a^2+b^2 $r^2=a^2+b^2$

Using trig ratios, we know that:

equation image indicator $LaTeX: \cos\theta=\frac{a}{r}\:and\:\sin\theta=\frac{b}{r}\\ r\cos\theta=a\:and\:r\sin\theta=b$ $\cos\theta=\frac{a}{r}\:and\:\sin\theta=\frac{b}{r}\\ r\cos\theta=a\:and\:r\sin\theta=b$

So, if we take our complex numbers and make some substitutions, we can determine that polar form of a complex number is:

equation image indicator $LaTeX: z=a+bi\:\Longrightarrow\:r\cos\theta+i\left(r\sin\theta\right)=r\left(\cos\theta+i\sin\theta\right)$ $z=a+bi\:\Longrightarrow\:r\cos\theta+i\left(r\sin\theta\right)=r\left(\cos\theta+i\sin\theta\right)$ . This final expression can be condensed to $LaTeX: r\:cis\:\theta.$ $r\:cis\:\theta.$

In addition, we can calculate θ by using the trig ratio: equation image indicator $LaTeX: \tan\theta=\frac{b}{a}$ $\tan\theta=\frac{b}{a}$ . When finding $LaTeX: \theta,\:$ $\theta,\:$ you will need to add 180° or π when a is negative. It is a good idea to plot the complex number and make sure $LaTeX: \theta\:$ $\theta\:$ lands in the correct quadrant.

While points in rectangular form and polar form are actually in the same location, we can use different types of graphs so that the points are easier to plot.

Rectangular Graph	Polar Graph

Let's use this information to write some complex numbers in polar form:

Write each complex number below in polar form.

Problem: $LaTeX: 2+2i\sqrt[]{3}$ $2+2i\sqrt[]{3}$ text annotation indicator

Solution: $LaTeX: 4\left(\cos60°\:+i\sin60°\right)$ $4\left(\cos60°\:+i\sin60°\right)$

Problem: $LaTeX: -\frac{3}{2}-\frac{3\sqrt[]{3}}{2}i$ $-\frac{3}{2}-\frac{3\sqrt[]{3}}{2}i$ text annotation indicator

Solution: $LaTeX: 3\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)$ $3\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)$

Problem: $LaTeX: -\frac{5}{2}+\frac{5\sqrt[]{3}}{2}i$ $-\frac{5}{2}+\frac{5\sqrt[]{3}}{2}i$ text annotation indicator

Solution: $LaTeX: 5\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$ $5\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$

Problem: $LaTeX: 3\sqrt[]{2}-3i\sqrt[]{2}$ $3\sqrt[]{2}-3i\sqrt[]{2}$ text annotation indicator

Solution: $LaTeX: 6\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)$ $6\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)$

Convert each polar form to complex numbers.

Problem: $LaTeX: 5\left(\cos300°+i\sin300°\right)$ $5\left(\cos300°+i\sin300°\right)$ text annotation indicator

Solution: $LaTeX: \frac{5}{2}-\frac{5\sqrt[]{3}}{2}i$ $\frac{5}{2}-\frac{5\sqrt[]{3}}{2}i$

Problem: $LaTeX: 5\left(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}\right)$ $5\left(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}\right)$ text annotation indicator

Solution: -5i

Problem: $LaTeX: 2\left(\cos60°+i\sin60°\right)$ $2\left(\cos60°+i\sin60°\right)$ text annotation indicator

Solution: $LaTeX: 1+i\sqrt[]{3}$ $1+i\sqrt[]{3}$

Problem: $LaTeX: 5\left(\cos30°+i\sin30°\right)$ $5\left(\cos30°+i\sin30°\right)$ text annotation indicator

Solution: $LaTeX: \frac{5\sqrt[]{3}}{2}+\frac{5}{2}i$ $\frac{5\sqrt[]{3}}{2}+\frac{5}{2}i$

At times, you might see the polar form of a complex number represented as $LaTeX: r\:cis\:\theta.$ $r\:cis\:\theta.$

The polar form of a complex number: equation image indicator $LaTeX: r\left(\cos\theta+i\sin\theta\right)\:\Longrightarrow\:r\:cis\:\theta$ $r\left(\cos\theta+i\sin\theta\right)\:\Longrightarrow\:r\:cis\:\theta$

So, let's try using that representation to graph some complex numbers on the polar plane.

IMAGES CREATED BY GAVS