Polar Coordinates

When you ask a child where a particular shop is, the child will answer roughly "just, over there" and point, describing the distance combined with the direction respectively. Or when you ask someone where their village is? They will answer "30 miles north of London" describing the distance and the direction rather than giving the latitude and longitude of their village.

In Mathematics, we have always been taught to represent the position of an object using cartesian coordinates which is not very natural or convenient. For a start, we should consider both the negative and the positive numbers to describe the points on the plane because using the direction and the distance as a means to describe the position is far more natural and convenient. 

Therefore, we can describe a polar coordinate system as a method in which a point is described by its distance from a fixed point at the center of the coordinate space known as a pole and by the measurement of the angle formed by a fixed-line and a line from the pole through the given point. In the polar coordinate system, the coordinates of a point are represented as (r, θ), where r is the distance of the point from the pole, and θ is the measure of the angle.

The polar coordinate system is just like an alternative to the Cartesian coordinate system. On one hand where the Cartesian system determines the position east and north of a fixed point. On the other hand, the polar coordinate system determines the location using direction and distance from a fixed point.

Polar Coordinates Formula

With the help of the formula, we can drive an infinite number of polar coordinates for just one coordinate point. The formula can be represented as:    

Where n is represented as an integer.

The value of θ will be positive if measured counterclockwise whereas it will be negative if measured clockwise. In the same way, the value of r will be positive if laid off the terminal side of θ whereas the value of r will be negative if laid off at the prolongation through the origin from the terminal side of θ. The side where the angle starts is known as the initial side whereas the ray where the measurement of the angle stops is known as the terminal side.

Plotting Points in Polar Coordinate System

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The two points are 3,60 and 4,210.

In a two-dimensional Polar Coordinate system, there are two polar coordinates: r and θ i.e, the radial coordinate which represents the radial distance from the pole and the angular coordinate which represents the anticlockwise angle from the 0° ray, respectively. It is also known as the positive x-axis on the Cartesian coordinate plane.

We can look at some polar coordinates examples for a better grasp. 

Consider that the polar coordinates (3,60°) are plotted as a point 3 units from the pole on the 60° ray. The coordinates (−3,240°) will also be plotted exactly at this point because the negative radial distance is measured as a positive distance on the opposite ray (240° − 180° = 60°).

Another important aspect of the Polar Coordinate System that is not present in the Cartesian coordinate system is the expressibility of a single point with an infinite number of different coordinates. Usually, the point (r, θ) can also be represented as (r, θ ± n × 360°) or (−r, θ ± (2n + 1)180°), 

where n is the integer. If the r coordinate of a point is 0, then the point will be located at the pole regardless of the θ coordinate.

Converting Cartesian Coordinate System to Polar Coordinate System 

If we know a point in Cartesian Coordinates (x,y) and want to convert it into Polar Coordinates (r,θ) we have to solve a right triangle with two known sides.

Example 1) What will be (12,5) in the Polar Coordinates system?

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Solution 1) We can use Pythagoras theorem to find the hypotenuse 

\[r^{2} = 12^{2} + 5^{2}\]

\[r = \sqrt{(12^{2} + 5^{2})}\]

\[r = \sqrt{(144 + 25)}\]

\[r = \sqrt{(169)}\]

r = 13.

Now, to find the angle, we will use the tangent function. 

Tan ( θ ) = 5 / 12 

θ = tan-1 ( 5 / 12 ) = 22.6°

Therefore, point (12,5) in the cartesian coordinate system will be (13, 22.6°) in the Polar Coordinate System.

Converting Polar Coordinate System to Cartesian coordinate System

Converting the polar coordinate system to Cartesian coordinate systems is relatively simple. We just have to take the cosine of θ in order to find the corresponding Cartesian x coordinate and sine of θ in order to find y.

Example 2) conversion from a polar coordinate system to the cartesian coordinate system. 

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Solution 2) With the help of basic trigonometry, it becomes easy to determine polar coordinates from a given pair of Cartesian coordinates.

\[r = \sqrt{x^{2} + y^{2}}\]

θ = tan-1(y/x)