# 4.1: Polar Coordinates

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

Everyone has dreamed of flying at one time or another. Not only would there be much less traffic to worry about, but directions would be so much simpler!

Walking or driving: "Go East 2 blocks, turn left, then North 6 blocks. Wait for the train. Turn right, East 3 more blocks, careful of the cow! Turn left, go North 4 more blocks and park."

Flying: "Fly 30deg East of North for a little less than 11 and 1/4 blocks. Land."

Nice daydream, what does it have to do with polar coordinates?

### Watch This

Embedded Video:

- Khan Academy: Polar Coordinates 1

### Guidance

The Polar Coordinate System is alternative to the Cartesian Coordinate system you have used in the past to graph functions. The polar coordinate system is specialized for visualizing and manipulating angles.

Angles are identified by travelling counter-clockwise around the circular graph from the 0deg line, or *r-axis* (where the + *x* axis would be) to a specified angle.

To plot a specific point, first go along the *r*-axis by *r* units. Then, rotate counterclockwise by the given angle, commonly represented "θ". Be careful to use the correct units for the angle measure (either radians or degrees).

*Radians*

Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used.

A **radian** is the angle formed between the *r* axis and a polar axis drawn to meet a section of the circumference that is the same length as the radius of a circle.

Given that the circumference of a circle is \begin{align*}2 \pi \cdot r\end{align*}, and since *r* is the radius, that means there are \begin{align*}2 \pi\end{align*} *radians* in a complete circle, and \begin{align*}1 \pi\end{align*} *radians* in 1/2 of a circle.

If 1/2 of a circle is \begin{align*}\pi\end{align*} radians, and is 180deg, that means that there are \begin{align*}\frac{180}{\pi}\end{align*} degrees in each *radian*.

That translates to approximately 57.3 degrees = 1 radian.

*Graphing using technology*

Polar Equations can be graphed using a graphing calculator: With the graphing calculator- go to **MODE**. There select **RADIAN** for the angle measure and **POL** (for Polar) on the **FUNC** (function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust *x*- and *y*- max/min, etc in **WINDOW.**

#### Example A

Plot the points on a polar coordinate graph:

Point A \begin{align*}\left (2,\frac{\pi}{3}\right )\end{align*}

Point B \begin{align*}(4, 135^{o})\end{align*}

Point C \begin{align*}\left (-2,\frac{\pi}{6}\right )\end{align*}

*Solution*

Below is the pole, polar axis and the points A, B and C.

#### Example B

Plot the following points:

a. \begin{align*}(4, 30^o)\end{align*}

b. \begin{align*}(2.5, \pi)\end{align*}

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

d. \begin{align*}\left(3,\frac{5\pi}{6}\right )\end{align*}

e. \begin{align*}(-2, 300^o)\end{align*}

*Solution*

#### Example C

Use a graphing calculator or plotting program to plot the following equations:

a. \begin{align*}r = 1 + 3 sin \theta\end{align*}

b. \begin{align*}r = 1 + 2 cos \theta\end{align*}

*Solution*

a.

b.

Review the steps above under ** Graphing using technology** if you are having trouble.

### Vocabulary

The ** polar coordinate system** is a specialized graph used for angles and angle manipulations.

The ** pole** is the center point on a polar graph.

One ** radian** is the angle formed by moving counter-clockwise around the circumference of a circle by the length of the radius. It is equal to apx 57.3 degrees.

The ** polar axis** is a ray drawn from the pole at the 0

*angle on a polar graph.*

^{o}### Guided Practice

1) Plot the points on a polar graph:

- a) \begin{align*}\left(2, \frac{\pi}{3}\right)\end{align*}
- b) \begin{align*}(3, 90^o)\end{align*}
- c) \begin{align*}(1.5, \pi)\end{align*}

2) Convert from radians to degrees:

- a) \begin{align*}\frac{\pi}{2}\end{align*}
- b) \begin{align*}5.17\end{align*}
- c) \begin{align*}\frac{3\pi}{2}\end{align*}

3) Convert from degrees to radians:

- a) \begin{align*}251^o\end{align*}
- b) \begin{align*}360^o\end{align*}
- c) \begin{align*}327^o\end{align*}

4) Convert from degrees to radians, answer in terms of \begin{align*}\pi\end{align*}:

- a) \begin{align*}90^o\end{align*}
- b) \begin{align*}270^o\end{align*}
- c) \begin{align*}45^o\end{align*}

*Answers*

1) The points are plotted on the graph below:

2) Recall that \begin{align*}\pi rad = 180^o\end{align*} and \begin{align*}1rad = \frac{180}{\pi} \approx 57.3^o\end{align*}

- a) If \begin{align*}\pi rad = 180^o\end{align*} then \begin{align*}\frac{\pi}{2}rad = 90^o\end{align*}
- b) If \begin{align*}1rad \approx 57.3^o\end{align*} then \begin{align*}5.17rad \approx 296^o\end{align*}
- c) If \begin{align*}\pi rad = 180^o\end{align*} then \begin{align*}\frac{3\pi}{2}rad = 270^o\end{align*}

3) Recall that \begin{align*}\frac{180^o}{\pi} = 57.3^o \approx 1rad\end{align*}

- a) If \begin{align*}57.3^o \approx 1rad\end{align*} then \begin{align*}251^o \approx 4.38rad \approx 1.4\pi rad\end{align*}
- b) If \begin{align*}57.3^o \approx 1rad\end{align*} then \begin{align*}360^o \approx 6.28rad\end{align*}
- c) If \begin{align*}57.3^o \approx 1rad\end{align*} then \begin{align*}\frac{327^o}{57.3^o} \approx 5.71 rad\end{align*}

4) Recall that \begin{align*}2\pi rad = 360^o\end{align*} and therefore \begin{align*}\pi rad = 180^o\end{align*}

- a) If \begin{align*}\pi rad = 180^o\end{align*} then \begin{align*}\frac{\pi}{2} rad = 90^o\end{align*}
- b) If \begin{align*} \pi rad = 180^o\end{align*} and \begin{align*}\frac{\pi}{2} rad = 90^o\end{align*} then \begin{align*}1\frac{1}{2}\pi rad \to \frac{3}{2}\pi \to \frac{3\pi}{2} rad = 270^o\end{align*}
- c) If \begin{align*}\frac{\pi}{2}rad = 90^o\end{align*} then \begin{align*}\frac{\pi}{4} rad = 45^o\end{align*}

### Practice

- Why can a point on the plane not be labeled using a unique ordered pair \begin{align*}(r, \theta)\end{align*}
- Explain how to graph \begin{align*}(r, \theta)\end{align*} if \begin{align*}r < 0\end{align*} and/or \begin{align*}\theta > 360\end{align*}

Graph Each Point in the Polar Plane

- A \begin{align*}(6, 145^o)\end{align*}
- B \begin{align*}\left(-2, \frac{13\pi}{6} \right)\end{align*}
- C \begin{align*}\left(\frac{7}{4}, -210^o\right)\end{align*}
- D \begin{align*}\left(5, \frac{\pi}{2}\right)\end{align*}
- E \begin{align*}\left(3.5, \frac{-\pi}{8}\right)\end{align*}

Name Two Other Pairs of Polar Coordinates for Each Point

- \begin{align*}(1.5, 170^o)\end{align*}
- \begin{align*}\left(-5, \frac{\pi}{-3}\right)\end{align*}
- \begin{align*}(3, 305^o)\end{align*}

Graph Each Polar Equation

- \begin{align*}r = 3\end{align*}
- \begin{align*}\theta = \frac{\pi}{5}\end{align*}
- \begin{align*}r = 15.5\end{align*}
- \begin{align*}r = 1.5\end{align*}
- \begin{align*}\theta = -175^o\end{align*}

Find the Distance Between Points

- \begin{align*}P_1 \left(5, \frac{\pi}{2}\right)\end{align*} and \begin{align*}P_2 \left(7, \frac{3\pi}{9}\right)\end{align*}
- \begin{align*}P_1 (1.3, -52^o) \end{align*} and \begin{align*}P_2 (-13.6, -162^o)\end{align*}
- \begin{align*}P_1 (3, 250^o) P_2 (7, 90^o)\end{align*}

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Cartesian coordinate system

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.polar axis

The polar axis is a ray drawn from the pole at the angle on a polar graph.polar coordinate system

The polar coordinate system is a special coordinate system in which the location of each point is determined by its distance from the pole and its angle with respect to the polar axis.pole

The pole is the center point on a polar graph.radian

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.### Image Attributions

Here you will learn about the polar coordinate system, which is similar in some ways to the (x, y) graphs you have worked with in the past, but is specialized for visually exploring angles.