CSPE - 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.

Convert Rectangular Equations to Polar Equations

Now that we can convert between polar coordinate and rectangular coordinates, we will learn to convert polar and rectangular equations. We will use the same procedure used to convert points between the coordinate systems. Let's take a look at the examples below.

Example: Write the rectangular equation in polar form. LaTeX: x^2+y^2=16 $x^2+y^2=16$

The goal is to eliminate x and y and reintroduce LaTeX: r $r$ and $LaTeX: \theta$ $\theta$ into the equation. In the previous examples above, we learned that $LaTeX: x=r\cos\theta$ $x=r\cos\theta$ and $LaTeX: y=r\sin\theta$ $y=r\sin\theta$ so we will substitute these into the equation.

$x^2+y^2=16$
$LaTeX: (r\cos\theta)^2+(r\sin\theta)^2=16$ $(r\cos\theta)^2+(r\sin\theta)^2=16$	Substitute $LaTeX: x=r\cos\theta$ $x=r\cos\theta$ and $LaTeX: y=r\sin\theta$ $y=r\sin\theta$
$LaTeX: r^2\cos^2\theta+r^2\sin^2\theta=16$ $r^2\cos^2\theta+r^2\sin^2\theta=16$	Simplify
$LaTeX: r^2(\cos^2\theta+\sin^2\theta)=16$ $r^2(\cos^2\theta+\sin^2\theta)=16$	Factor out $r^2$
$r^2(1)=16$	Use the Pythagorean Identity to substitute $LaTeX: \cos^2\theta+\sin^2\theta=1$ $\cos^2\theta+\sin^2\theta=1$
$LaTeX: r=\pm4$ $r=\pm4$	Square root both sides of the equation.

We can graph both equations and see that they are the same.

$x^2+y^2=16$	$r=4$ and $r=-4$

Let's try a different example.

Rewrite LaTeX: x^2+y^2=3y $x^2+y^2=3y$ as a polar equation.

This example does not have variable r so we must reintroduce it along with $LaTeX: \theta$ $\theta$ . The standard equation in rectangular form is LaTeX: x^2+y^2=r^2 $x^2+y^2=r^2$ so we will use this to substitute.

$x^2+y^2=3y$
$r^2=3y$	Since $x^2+y^2=r^2$ , we can substitute $r^2=3y$
$LaTeX: r^2=3r\sin\theta$ $r^2=3r\sin\theta$	Substitute $LaTeX: y=r\sin\theta$ $y=r\sin\theta$
$LaTeX: r^2-3r\sin\theta=0$ $r^2-3r\sin\theta=0$	Subtract $LaTeX: 3r\sin\theta$ $3r\sin\theta$ on both sides to set the equation equal to zero.
$LaTeX: r(r-3\sin\theta)=0$ $r(r-3\sin\theta)=0$	Factor out and solve for r.
$r=0$ and $LaTeX: r=3\sin\theta$ $r=3\sin\theta$	$r=0$ represents the point (0, 0) so this is not the solution. Therefore, $LaTeX: r=3\sin\theta$ $r=3\sin\theta$ is the polar equation.

We can see that the graphs of both equations are the same.

$x^2+y^2=3y$	$LaTeX: r=3\sin\theta$ $r=3\sin\theta$

Time to practice on your own! Try these problems below and check your answers with the solutions shown.

Convert Polar Equations to Rectangular Equations

Now let's learn how to convert polar equations to rectangular equations! Here is an example.

Rewrite $LaTeX: r=4\cos\theta$ $r=4\cos\theta$ in rectangular form. Remember that the equation in rectangular form is LaTeX: x^2+y^2=r^2 $x^2+y^2=r^2$ . Our goal is is to eliminate r and $LaTeX: \theta$ $\theta$ and reintroduce x and y into the equation.

$LaTeX: r=4\cos\theta$ $r=4\cos\theta$
$LaTeX: r^2=4r\cos\theta$ $r^2=4r\cos\theta$	Multiply both sides by r.
$LaTeX: x^2+y^2=4r\cos\theta$ $x^2+y^2=4r\cos\theta$	Substitute this into the equation $x^2+y^2=r^2$ .
$x^2+y^2=4x$	Recall that $LaTeX: x=r\cos\theta$ $x=r\cos\theta$
$x^2-4x+y^2=0$	Subtract 4x on both sides to set the equation equal to zero.
$x^2-4x+4+y^2=4$ $(x-2)^2+y^2=4$	Complete the square to rewrite the equation in standard form.

Your turn to try! Complete these practice problems below.

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