# 6.1: Polar Coordinates

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

## Introduction

This chapter introduces and explores the polar coordinate system, which is based on a radius and theta. Students will learn how to plot points and basic graphs in this system as well as convert \begin{align*}x\end{align*}

## Learning Objectives

- Distinguish between and understand the difference between a rectangular coordinate system and a polar coordinate system.
- Plot points with polar coordinates on a polar plane.

## Plotting Polar Coordinates

The graph paper that you have used for plotting points and sketching graphs has been rectangular grid paper. All points were plotted in a rectangular form \begin{align*}(x, y)\end{align*}

Look at the two options below:

You are all familiar with the rectangular grid paper shown above. However, the circular paper lends itself to new discoveries. The paper consists of a series of concentric circles-circles that share a common center. The common center \begin{align*}O\end{align*}

These coordinates are the result of assuming that the angle is rotated counterclockwise. If the angle were rotated clockwise then the coordinates of \begin{align*}P\end{align*}

**Example 1:** Plot the point \begin{align*}A (5, -255^\circ)\end{align*}

**Solution, A:** To plot \begin{align*}A\end{align*}**clockwise** from the polar axis and plot the point on the circle. Label it \begin{align*}A\end{align*}

**Solution, B:** To plot \begin{align*}B\end{align*}**counter clockwise** from the polar axis and plot the point on the circle. Label it \begin{align*}B\end{align*}

These points that you have plotted have \begin{align*}r\end{align*}

**The point is reflected across the pole to point \begin{align*}M\end{align*}.**

There are multiple representations for the coordinates of a polar point \begin{align*}P(r, \theta)\end{align*}. If the point \begin{align*}P\end{align*} has polar coordinates \begin{align*}(r, \theta)\end{align*}, then \begin{align*}P\end{align*} can also be represented by polar coordinates \begin{align*}(r, \theta + 360k^\circ)\end{align*} or \begin{align*}(-r, \theta + [2k + 1] 180^\circ)\end{align*} if \begin{align*}\theta\end{align*} is measured in degrees or by \begin{align*}(r, \theta + 2 \pi k)\end{align*} or \begin{align*}(-r, \theta + [2k + 1] \pi)\end{align*} if \begin{align*}\theta\end{align*} is measured in radians. Remember that \begin{align*}k\end{align*} is any integer and represents the number of rotations around the pole. Unless there is a restriction placed upon \begin{align*}\theta\end{align*}, there will be an infinite number of polar coordinates for \begin{align*}P(r, \theta)\end{align*}.

**Example 2:** Determine four pairs of polar coordinates that represent the following point \begin{align*}P(r, \theta)\end{align*} such that \begin{align*}-360^\circ \le \theta \le 360^\circ\end{align*}.

**Solution:** Pair 1 \begin{align*}\rightarrow (4, 120^\circ)\end{align*}. Pair 2 \begin{align*}\rightarrow (4, -240^\circ)\end{align*} comes from using \begin{align*}k = -1\end{align*} and \begin{align*}(r, \theta + 360^\circ k), (4, 120^\circ + 360(-1))\end{align*}. Pair 3 \begin{align*}\rightarrow (-4, 300^\circ)\end{align*} comes from using \begin{align*}k = 0\end{align*} and \begin{align*}(-r, \theta + [2k + 1] 180^\circ), (-4, 120^\circ + [2(0) + 1] 180^\circ)\end{align*}. Pair 4 \begin{align*}\rightarrow (-4, -60^\circ)\end{align*} comes from using \begin{align*}k = -1\end{align*} and \begin{align*}(-r, \theta + [2k + 1] 180^\circ), (-4, 120^\circ + [2(-1) + 1] 180^\circ)\end{align*}.

These four pairs of polar coordinates all represent the same point \begin{align*}P\end{align*}. You can apply the same procedure to determine polar coordinates of points that have \begin{align*}\theta\end{align*} measured in radians. This will be an exercise for you to do at the end of the lesson.

## The Distance between Two Polar Coordinates

Just like the Distance Formula for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates, there is a way to find the distance between two polar coordinates. One way that we know how to find distance, or length, is the Law of Cosines, \begin{align*}a^2 = b^2 + c^2 - 2bc \cos A\end{align*} or \begin{align*}a = \sqrt{b^2 + c^2 - 2bc \cos A}\end{align*}. If we have two points \begin{align*}(r_1, \theta_1)\end{align*} and \begin{align*}(r_2, \theta_2)\end{align*}, we can easily substitute \begin{align*}r_1\end{align*} for \begin{align*}b\end{align*} and \begin{align*}r_2\end{align*} for \begin{align*}c\end{align*}. As for \begin{align*}A\end{align*}, it needs to be the angle between the two radii, or \begin{align*}(\theta_2 - \theta_1)\end{align*}. Finally, \begin{align*}a\end{align*} is now distance and you have \begin{align*}d = \sqrt{r^2_1 + r^2_2 - 2 r_1 r_2 \cos (\theta_2 - \theta_1)}\end{align*}.

**Example 3:** Find the distance between \begin{align*}(3, 60^\circ)\end{align*} and \begin{align*}(5, 145^\circ)\end{align*}.

**Solution:** After graphing these two points, we have a triangle. Using the new Polar Distance Formula, we have \begin{align*}d = \sqrt{3^2 + 5^2 - 2(3)(5) \cos 85^\circ} \approx 5.6\end{align*}.

**Example 4:** Find the distance between \begin{align*}(9, -45^\circ)\end{align*} and \begin{align*}(-4, 70^\circ)\end{align*}.

**Solution:** This one is a little trickier than the last example because we have negatives. The first point would be plotted in the fourth quadrant and is equivalent to \begin{align*}(9, 315^\circ)\end{align*}. The second point would be \begin{align*}(4, 70^\circ)\end{align*} reflected across the pole, or \begin{align*}(4, 250^\circ)\end{align*}. Use these two values of \begin{align*}\theta\end{align*} for the formula. Also, the radii should always be positive when put into the formula. That being said, the distance is \begin{align*}d = \sqrt{9^2 + 4^2 - 2(9)(4) \cos (315-250)^\circ} \approx 8.16\end{align*}.

## Points to Consider

- How is the polar coordinate system similar/different from the rectangular coordinate system?
- How do you plot a point on a polar coordinate grid?
- How do you determine the coordinates of a point on a polar grid?
- How do you calculate the distance between two points that have polar coordinates?

## Review Questions

- Graph each point:
- \begin{align*}M (2.5, 210^\circ)\end{align*}
- \begin{align*}S \left (-3.5, \frac{5 \pi}{6} \right )\end{align*}
- \begin{align*}A \left (1, \frac{3 \pi}{4} \right )\end{align*}
- \begin{align*}Y \left (5.25, - \frac{\pi}{3} \right )\end{align*}

- For the given point \begin{align*}A \left (-4, \frac{\pi}{4} \right )\end{align*}, list three different pairs of polar coordinates that represent this point such that \begin{align*}-2\pi \le \theta \le 2\pi\end{align*}.
- For the given point \begin{align*}B (2, 120^\circ)\end{align*}, list three different pairs of polar coordinates that represent this point such that \begin{align*}-2\pi < \theta < 2 \pi\end{align*}.
- Given \begin{align*}P_1\end{align*} and \begin{align*}P_2\end{align*}, calculate the distance between the points.
- \begin{align*}P_1 (1,30^\circ)\end{align*} and \begin{align*}P_2 (6,135^\circ)\end{align*}
- \begin{align*}P_1 (2,-65^\circ)\end{align*} and \begin{align*}P_2 (9,85^\circ)\end{align*}
- \begin{align*}P_1 (-3,142^\circ)\end{align*} and \begin{align*}P_2 (10,-88^\circ)\end{align*}
- \begin{align*}P_1 (5,-160^\circ)\end{align*} and \begin{align*}P_2 (16, -335^\circ)\end{align*}

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