# 6.8: Equivalent Polar Curves

**At Grade**Created by: CK-12

**Practice**Equivalent Polar Curves

While working on a problem in math class, you get a solution with a certain equation. In this case, your solution is \begin{align*}3 + 2\cos (\theta)\end{align*}

### Equivalent Polar Curves

The expression “same only different” comes into play in this section. We will graph two distinct polar equations that will produce two equivalent graphs. Use your graphing calculator and create these curves as the equations are presented.

In some other section, graphs were generated of a limaçon, a dimpled limaçon, a looped limaçon and a cardioid. All of these were of the form \begin{align*}r = a \pm b \sin \theta\end{align*}

#### Comparing Graphs of Polar Equations

Plot the following polar equations and compare the graphs.

\begin{align*}r & = 1 + 2 \sin \theta \\
r & = -1 + 2 \sin \theta\end{align*}

By looking at the graphs, the result is the same. So, even though \begin{align*}a\end{align*}

\begin{align*}r & = 4 \cos \theta \\
r & = 4 \cos (-\theta)\end{align*}

These functions also result in the same graph. Here, \begin{align*}\theta \end{align*}

#### Describing Graphs

1. Graph the equations \begin{align*}x^2 + y^2 = 16\end{align*}

Both equations, one in rectangular form and one in polar form, are circles with a radius of 4 and center at the origin.

2. Graph the equations \begin{align*}(x - 2)^2 + (y + 2)^2 = 8\end{align*}

There is not a visual representation shown here, but on your calculator you should see that the graphs are circles centered at (2, -2) with a radius \begin{align*}2 \sqrt{2} \approx 2.8\end{align*}

### Examples

#### Example 1

Earlier, you were asked if there is a way you can determine if the two equations are equivalent.

As you learned in this section, we can compare graphs of equations to see if the equations are the same or not.

A graph of \begin{align*}3 + 2\cos (\theta)\end{align*}

And a graph of \begin{align*}-3 + 2\cos (-\theta)\end{align*}

As you can see from the plots, your friend is correct. Your graph and his are the same, therefore the equations are equivalent.

#### Example 2

Write the rectangular equation \begin{align*}x^2 + y^2 = 6x\end{align*}

\begin{align*}x^2 + y^2 & = 6x \\
r^2 & = 6(r \cos \theta) && r^2 = x^2 + y^2 \qquad \ \text{and} \qquad \ x = y \cos \theta \\
r & = 6 \cos \theta && \text{divide by}\ r \end{align*}

Both equations produced a circle with center \begin{align*}(3, 0)\end{align*}

#### Example 3

Determine if \begin{align*}r = -2 + \sin \theta\end{align*}*without* graphing.

\begin{align*}r = - 2 + \sin \theta\end{align*}

#### Example 4

Determine if \begin{align*}r = -3 + 4 \cos (- \pi)\end{align*}*without* graphing.

\begin{align*}r = -3 + 4 \cos ( - \pi)\end{align*}

### Review

For each equation in rectangular form given below, write the equivalent equation in polar form.

- \begin{align*}x^2+y^2=4\end{align*}
x2+y2=4 - \begin{align*}x^2+y^2=6y\end{align*}
x2+y2=6y - \begin{align*}(x-1)^2+y^2=1\end{align*}
(x−1)2+y2=1 - \begin{align*}(x-4)^2+(y-1)^2=17\end{align*}
(x−4)2+(y−1)2=17 - \begin{align*}x^2+y^2=9\end{align*}
x2+y2=9

For each equation below in polar form, write another equation in polar form that will produce the same graph.

- \begin{align*}r=4+3\sin\theta\end{align*}
r=4+3sinθ - \begin{align*}r=2-\sin\theta\end{align*}
- \begin{align*}r=2+2\cos\theta\end{align*}
- \begin{align*}r=3-\cos\theta\end{align*}
- \begin{align*}r=2+\sin\theta\end{align*}

Determine whether each of the following sets of equations produce equivalent graphs *without* graphing.

- \begin{align*}r=3-\sin \theta\end{align*} and \begin{align*}r=3+\sin \theta\end{align*}
- \begin{align*}r=1+2\sin \theta\end{align*} and \begin{align*}r=-1+2\sin \theta\end{align*}
- \begin{align*}r=3\sin \theta\end{align*} and \begin{align*}r=3\sin ( -\theta )\end{align*}
- \begin{align*}r=2\cos \theta\end{align*} and \begin{align*}r=2\cos (-\theta )\end{align*}
- \begin{align*}r=1+3\cos \theta\end{align*} and \begin{align*}r=1-3\cos \theta\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.8.

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### Image Attributions

Here you'll learn to determine if two polar equations are equivalent by inspection of their respective graphs.

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