# 6.1: Plots of Polar Coordinates

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

**Practice**Plots of Polar Coordinates

While playing a game of darts with your friend, you decide to see if you can plot the coordinates of where your darts land. The dartboard looks like this

While trying to set up a rectangular coordinate system, your friend tells you that it would be easier to plot the positions of your darts using a "polar coordinate system". Can you do this?

At the end of this Concept, you'll be able to accomplish this plotting task successfully.

### Watch This

James Sousa: Introduction to Polar Coordinates

### Guidance

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 A

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*}

#### Example B

Determine four pairs of polar coordinates that represent the following point \begin{align*}P(r, \theta)\end{align*}

**Solution:** Pair 1 \begin{align*}\rightarrow (4, 120^\circ).\end{align*}

#### Example C

Plot the following coordinates in polar form and give their description in polar terms: (1,0), (0,1), (-1,0), (-1,1)

**Solution:** The points plotted are shown above. Since each point is 1 unit away from the origin, we know that the radius of each point in polar form will be equal to 1.

The first point lies on the positive 'x' axis, so the angle in polar coordinates is \begin{align*}0^\circ\end{align*}

### Vocabulary

**Polar Coordinates:** A set of ** polar coordinates** are a set of coordinates plotted on a system that uses the distance from the origin and angle from an axis to describe location.

### Guided Practice

1. Plot the point: \begin{align*}M (2.5, 210^\circ)\end{align*}

2. \begin{align*}S \left (-3.5, \frac{5 \pi}{6} \right )\end{align*}

3. \begin{align*}A \left (1, \frac{3 \pi}{4} \right )\end{align*}

**Solutions:**

1.

2.

3.

### Concept Problem Solution

Since you have the positions of the darts on the board with both the distance from the origin and the angle they make with the horizontal, you can describe them using polar coordinates.

As you can see, the positions of the darts are:

\begin{align*}\left( 3,45^\circ \right)\end{align*}, \begin{align*}\left( 6, 90^\circ \right)\end{align*}

and

\begin{align*}\left( 4,0^\circ \right)\end{align*}

### Practice

Plot the following points on a polar coordinate grid.

- \begin{align*}(3, 150^\circ )\end{align*}
- \begin{align*}(2, 90^\circ )\end{align*}
- \begin{align*}(5, 60^\circ )\end{align*}
- \begin{align*}(4, 120^\circ )\end{align*}
- \begin{align*}(3, 210^\circ )\end{align*}
- \begin{align*}(-2, 120^\circ )\end{align*}
- \begin{align*}(4, -90^\circ )\end{align*}
- \begin{align*}(-5, -30^\circ )\end{align*}
- \begin{align*}(2, -150^\circ )\end{align*}
- \begin{align*}(-3, 300^\circ )\end{align*}

Give three alternate sets of coordinates for the given point within the range \begin{align*}-360^\circ \leq\theta\leq 360^\circ\end{align*}.

- \begin{align*}(3, 60^\circ )\end{align*}
- \begin{align*}(2, 210^\circ )\end{align*}
- \begin{align*}(4, 330^\circ )\end{align*}
- Find the length of the arc between the points \begin{align*}(2, 30^\circ )\end{align*} and \begin{align*}(2, 90^\circ )\end{align*}.
- Find the area of the sector created by the origin and the points \begin{align*}(4, 30^\circ ) \end{align*} and \begin{align*}(4, 90^\circ ) \end{align*}.

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### Image Attributions

Here you'll learn how to express a point in a polar coordinate system as a distance from an origin and the angle with respect to an axis.