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# Transformation from Rectangular to Polar Coordinates

A plane can be defined as a point having zero-dimensions or a line having one-dimension or a space which encompasses three-dimensions. Any Point taken on this plane can be represented in two different ways that are:

1. Rectangular coordinates and
2. Polar coordinates.
The Rectangular coordinates are present in the form of P(x, y), where 'x' and 'y' are the horizontal and vertical distances from the origin.
The Polar coordinate system is defined in terms of distance from a fixed point and an angle when viewed from a particular direction. Let the distance of a point P(x, y) from origin (an arbitrary fixed point) be denoted by ‘h’ denoted by the symbol O). Consider the angle between the radial line from the point P to O and the given line “θ = 0” (a kind of positive axis for our polar coordinate system be angle ‘A’. Where,
h ≥ 0  &  0≤ A < 2π.
The transformation of Rectangular Co-ordinates to the Polar co-ordinates can be done using certain formulae:

In the given diagram we have,
By the rule of Pythagoras:
r = √ (x2 + y2)
And the Slope can be found by tan A,
or tan  A = y / x  ,  so therefore:
A = tan-1 (y / x)
So the rectangular point: (x, y) can be converted to polar coordinates like this:
(√ (x2 + y2), tan-1 (y / x)).
An Example can be taken to understand this transformation better,
Example: A point is having rectangular coordinates as (3, 4). Find out the corresponding Polar Coordinates.
Solution:  r = radius or distance of the point from the Centre (fixed point) = square root of
(32 + 42) = 5, and
Angle   A = tan-1(4/3) = 53.13º
so, the polar co-ordinates can be given as
(r, A) = (5, 53.13º).