Polar Coordinate System

The polar coordinate system makes many problems in mathematics easy to solve. Learn about this tool in this wonderful mathematics tool in an understandable article.

Polar Coordinates

Polar coordinates are most useful when it is convenient to know distance and angle. In many such cases using Cartesian coordinates is cumbersome.

Polar coordinates use the distance from the origin, called the radius, and the angle from the positive horizontal axis in the counterclockwise direction to locate a point.

A point a distance r from the origin and an angle θ counterclockwise from the positive axis is identified (r, θ) in the polar coordinate system.

Cartesian Coordinates to Polar Coordinates

If starting from the Cartesian coordinate system to convert to the polar coordinate system use the Pythagorean theorem, x² + y² = r². The angle is found by using the arctangent of the angle at the origin, tan^-1(y/x).

The above leads to the equations r = (x² + y²)^1/2 and θ = tan^-1(y/x). Points where x = 0 must be handled with thought since the arctangent is not defined. In such cases θ = π/2 when y is positive, and θ = 3π/2 when y is negative. Also, the signs of both x and y need be considered, and adjustments made if the point (x, y) is not in the first quadrant.

Example 1: Find the polar coordinate representation of the point (3, 4).

r = (x² + y²)^1/2 = (3² + 4²)^1/2 = (9 + 16)^1/2 = 5

θ = tan^-1(4/3)

(5, tan^-1(4/3))

Example 2: Find the polar coordinate representation of the point (-3, -3).

In this example there are two ways to approach the transformation into polar coordinates.

In one case the radius is -3, taking the negative root of ((-3)² + (-3)²)^1/2 = -3 and θ = tan^-1(1) = π/4.

Another possibility is ((-3)² + (-3)²)^1/2= 3 and θ = tan^-1(1) = π/4, adjusted to 5π/4 since both x and y are negative, placing the point in the third quadrant.

(-3, π/4) or equivalently (3, 5π/4)

Example 3: Find the polar coordinate representation of the point (0, 0).

The radius is 0, and the angle is arbitrary.

Example 4: Find the polar coordinate representation of the point (0, 3).

The radius is (0² + 3²)^1/2 = 3, and θ = π/2, since y is positive. The arctangent does not exist for any angle terminating of the vertical axis.

Polar Coordinates to Cartesian Coordinates

Basic trigonometry gives the definitions of cos(θ) as x/r and sin(θ) as y/r. Solving cos(θ) = x/r for x gives x = r cos(θ). Solving sin(θ) = y/r for x gives y = r sin(θ).

Example 1: Find the Cartesian coordinate representation of (r, θ) = (2, π/3).

x = r cosθ = 2(1/2) = 1

y = r sinθ = 2(2^1/2/2) = 2^1/2

(1, 2^1/2)

Convert Equations in Cartesian Coordinates to Polar Coordinates

It is possible to convert entire equations from Cartesian coordinates to polar coordinates. Simply replace x with r cosθ and y with r sinθ.

Example 1: Transform x² + y² = 3x into polar coordinates.

(r cosθ)² + (r sinθ)² = 3r cosθ

r²cos²θ + r²sin²θ = 3r cosθ

r² = 3r cosθ

r = 3 cosθ

Convert Equations in Polar Coordinates to Cartesian Coordinates

To convert an equation in polar coordinates to Cartesian coordinates replace r with (x² + y²)^1/2 and θ with arctan(y/x), which can be awkward. It may be better to eliminate θ using x = r cosθ or y = r sinθ. Notice any of the above expressions for θ can be used in the same problem. In fact, it is allowed to express any trigonometry function in terms of x, y, and r = (x² + y²)^1/2.

Example 1: Express r² = cotθ in Cartesian coordinates.

r² = x² + y² and cotθ = x/y.

r² = x/y

Polar Coordinate Graph Paper

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Polar Graphs

The graph of θ = a is a straight line.

The graph of r = aθ is a spiral.

The graph of r = a is a circle with a radius of a centered at the origin.

The graph of r = a cosθ is a circle of diameter a that goes from (0, 0) to (0, a).

The graph of r = a sinθ is a circle of diameter a that goes from (0, 0) to (a, 0).

A limaçon has a graph symmetric about the horizontal axis when an equation is of the form r = b + a cosθ and a graph symmetric about θ = π/2 for an equation of the form r = b + a sinθ.

Notice negative values can occur for b. If a is negative factor our -1. If b is 0 the graph is a circle, as handled above.

There are four cases for limaçons.

If the magnitude of a equals the magnitude of b the graph is a cardioid with a sharp point.

If the magnitude of a is less than the magnitude of b there is an inner loop.

If the magnitude of a is between the magnitude of b and of 2b there is a dimple.

If the magnitude of a exceeds the magnitude of 2b, the figure is convex. A figure is convex means that a line segment can be drawn between any two points enclosed by the figure and the entire line segment is within the figure.

A rose figure results from the graph of an equation of the form r = a cosnθ or of the form a sinnθ. The figure had n petals if n is odd and 2n petals if n is even. The reason for the difference is that when n is odd the graph repeats itself.

Polar Coordinate Dry Erase

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Graphing Calculator

Using a graphing calculator for polar coordinates is tricky. If Δθ is selected too large the graph can be unrecognizable. The reason is graphing calculators determine points, then connect those points with line segments. Graphing a rose, for example, can have the graph jump from one petal to another with too large of an increment for θ.

TI-84

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Extension to Cylindrical Coordinates and to Spherical Coordinates

Both cylindrical and spherical coordinate systems use polar coordinates with an added dimension. For cylindrical coordinates the added dimension is z. For spherical coordinates the third dimension is an angle..