# Introduction to Polar Coordinates

## What is Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system used to locate the position of a point by how far it is from a reference point and its angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis.

The position of any point in the polar coordinate system can be referenced by the polar coordinates (r,θ). The first coordinate (r), commonly referred to as the radial coordinate or radius, indicates how far a point is from the pole. The angle (θ), commonly referred to as the angular coordinate or azimuth, indicates the angle between the radius (r) and the polar axis.

By convention, the reference direction (polar axis) is usually drawn as a horizontal ray from the pole to the right, and the polar angle increases to positive for counterclockwise rotations. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°).

The polar coordinate system can be extended to three dimensions in two ways: the cylindrical and spherical coordinate systems.

## Plotting Points Using Polar Coordinates

Plotting points on a polar graph is a straightforward task. First, we measure an angle equal to (θ) from the reference line (polar axis), and then we draw a line of length (r) from the pole in that direction. For example, the point (4, 60°) is plotted by rotating 60° degrees counterclockwise from the polar axis, then moving a distance of 4 units from the pole in that direction.

But what if the radius (r) coordinate were negative? Well, the easiest way to plot a point with a negative (r) component is to rotate the given angle (θ) from the polar axis and draw (r) as if it were positive in that direction, then take the mirror of (r) or (-r).

By convention counterclockwise direction is considered positive, thus a negative θ simply means we move in the opposite (clockwise) direction.

### Plotting Points Using Polar Coordinates Example 1.1

Try to plot the point (6, 45°) on a polar graph.

Step 1) We identify the r and θ coordinates as:

r-coordinate = 6 θ-coordinate = 45°Step 2) We rotate 45° degrees from the polar axis in a counterclockwise direction, as shown in the diagram below.

Step 3) Now, we draw a line of 6 units-length from the pole in that direction. The point where the line ends is the point with coordinates (6,45°). We plot the point there, as shown in the diagram below.

We should note that adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with a negative radial component and an opposite direction (adding 180° to the angular coordinate). Therefore, the same point (r, θ) can be expressed with an infinite number of different polar coordinates (r, θ + n × 360°) and (−r, θ + 180° + n × 360°) = (−r, θ + (2n + 1) × 180°), where n is an arbitrary integer.

## Converting Between Polar And Cartesian Coordinates

### 1) How to convert from polar to cartesian coordinates

When given a set of polar coordinates (r,θ), we may need to convert them to cartesian coordinates x and y. To do so, we can use the trigonometric functions sine and cosine:

Dropping a perpendicular line from the point in the plane to the x-axis forms a right triangle, as shown in the diagram below. An easy way to remember the equations above is to think of cos(θ) as the adjacent side over the hypotenuse and sin(θ) as the opposite side over the hypotenuse.

### Converting Polar to Cartesian Coordinates Example 2.1

Convert the polar coordinates (3, 90°) to cartesian coordinates x and y.

Step 1) We identify the r and θ coordinates as:

r-coordinate = 3 θ-coordinate = 90°Step 2) We Use the equivalent relationships:

x = r cos(θ) = 3 cos(90°) = 0 y = r sin(θ) = 3 sin(90°) = 3The cartesian coordintes are **(0,3)**.

Converting Polar to Cartesian Coordinates Example 2.2

Convert the polar coordinates (-2, 0°) to cartesian coordinates x and y.

Step 1) We identify the r and θ coordinates as:

r-coordinate = -2 θ-coordinate = 0°Step 2) We Use the equivalent relationships:

x = r cos(θ) = -2 cos(0°) = -2 y = r sin(θ) = -2 sin(0°) = 0The cartesian coordintes are also **(-2,0)**.

### 2) How to convert from cartesian to polar coordinates

The cartesian coordinates x and y can be converted to polar coordinates r and θ with r ≥ 0 and θ in the interval (0, 2π) by:

π is equal to 180°.

Converting Cartesian to Polar Coordinates Example 2.3

Convert the cartesian coordinates (3,3) to polar coordinates (r, θ).

Step 1) Since point (3,3) is located in the first quadrant (i.e. both x and y are positive), we can find θ as:

θ = arctan(y/x) θ = arctan(3/3) = 45°Step 2) Now, we find r using the Pythagorean Theorem as:

So, the polar coordinates of the point are **(4.2, 45°)**.

There are other sets of polar coordinates that will be the same as our solution. For example, the points (4.2,405°) and (-4.2, 225°) will coincide with the original solution of (4.2,45°).

### Frequently Asked Questions

- Why polar coordinates were invented?
- The initial motivation for the polar system was the study of circular and orbital motion.

- Where are polar coordinates used?
- Polar coordinates are used in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Navigation systems are a good real-life application of polar coordinates.

- What is the 3d version of the polar coordinate system?
- The cylindrical and spherical coordinate systems are extensions of the polar system in 3d.

**References:**Jay Abramson. “Precalculus 2e - 8.3 Polar Coordinates.” OpenStax, 2021. December 21. https://openstax.org/books/precalculus-2e/pages/8-3-polar-coordinates.