polar coordinates

Polar coordinate is a way to represent a point in space.

A- Pole

B- Polar Axis

r- Radial Coordinate

θ - Angular Coordinate

A polar coordinate system refers to a two-dimensional coordinate system in which every point on a plane is influenced by a distance from a point of reference and an angle from a direction of reference. So in Polar coordinates, we talk about the distance of the point from pole A which is represented here as r, and the angle that it makes with the axis to reach to point D which is θ here.

The reference point (which is analogous to the origin of a Cartesian coordinate system) is referred to as the pole. The ray from the pole in the direction of reference is what is known as the polar axis. Radius, radial distance, or radial coordinate is the distance from the pole. The angle from the pole is referred to as the azimuth, polar angle, or angular coordinate.

 

To define a point in polar coordinates, we use the notation (r, θ), where r is the distance from the origin to the point, and θ is the angle measured counterclockwise from the polar axis to the line connecting the origin to the point.

To convert between polar and rectangular coordinates (the traditional (x, y) system), we use the following formulas:

\(x = r \times \cosθ\)

\(y = r \times \sinθ\)

Conversely, to convert from rectangular coordinates to polar coordinates, we use the formulas:

\(r = \sqrt{x^2 + y^2}\)

\(θ = \tan^{-1}{\frac{y}{x}}\)

Here, tan^-1 is a function that takes two arguments (y and x) and returns the angle between the positive x-axis and the line passing through the origin and (x, y), measured counterclockwise.

Polar coordinates are particularly useful in situations involving circular or rotational symmetry, such as describing the positions of points on a wheel or the movement of a pendulum. They are also used extensively in physics and engineering to describe the motion of objects in polar coordinates, such as satellites orbiting the earth.

CONVERTING

To convert one from a polar coordinate to a Cartesian coordinate or vice versa, we use a triangle.

CONVERTING FROM CARTESIAN TO POLAR

In case we know a point in Cartesian Coordinates (x, y) and we wish to convert it into polar coordinates (r, θ), we start by solving a right triangle with two sides that are known.

For example, What is (12, 5) in polar Coordinates?

SOLUTION

Use Pythagoras Theorem to find the long side (hypotenuse).

r² = 12² + 5²

r² = 169

r = 13

The second step is to use the tangent function to find the angle,

\(\tanθ = {5 \over 12}\)

\(θ = \tan^{-1} {\frac{5}{12}} = 22.6⁰ \)(to one decimal point).

Answer: The point (12,5) is (13, 22.6°) in Polar Coordinates.

CONVERTING FROM POLAR TO CARTESIAN

In case we know a point in Polar Coordinates (r, θ), and we wish to convert it to Cartesian Coordinates (x,y), we solve a right triangle with a known hypotenuse and angle.

For example: convert (13, 22.6°) to Cartesian Coordinates.

SOLUTION

Use the Cosine Function for x: \(\cos 22.6⁰ = \frac{x}{13}\)

x = \(13 \times \cos 22.6⁰\)

x = 13 × 0.923

x = 12.002

x = 12 (round off)

Use the Sine Function for y: \(\sin 22.6⁰ = \frac{y}{13}\)

y = 13 × \(\sin 22.6⁰\)

y = 13 × 0.391

y = 4.996

y = 5 (round off)

Answer: The point (13, 22.6°) is (12, 5) in Cartesian Coordinates.