Coordinates Systems

In physics we always need to find the locations of objects in space. The mathematical description of an object's motion requires a method for describing the object's position at various times.
Fig. 1: Cartesian Coordinate System
Fig. 2: The plane polar coordinates of a point are represented by the distance r and the angle $\theta$
This description is accomplished with the use of coordinates system, where one of the system is a well known system called Cartesian Coordinate System, in which horizontal and vertical axes intersect at a point taken to be the origin.
Objects can be shown in the space with a position referenced to the origin of the system. Cartesian coordinates are also called rectangular coordinates.
Sometimes it is more convenient to represent a point in a plane by its plane polar coordinates ($r, \theta $), as shown in Figure 2. In this polar coordinate system, r is the distance from the origin to the point having Cartesian Coordinates (x, y), and $\theta$ is the angle between r and a fixed axis. This fixed axis is usually the positive x axis, and $\theta$ is usually measured counterclockwise from it.
Fig. 3: The right triangle used to relate (x, y) to ($r, \theta $).
If we draw a perpendicular line from point to the x-axis, we will get the right triangle as show in Figure 3. By having an idea of Trigonometric functions, we can relate the Cartesian Coordinates with Polar System by applying trigonometric functions.
we find that
$$\sin\theta = {y \over r}$$
$$\cos\theta = {x \over r}$$
Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian Coordinates, using the equations given below.
$$ x = r \cos(\theta)$$
$$ y = r \sin(\theta)$$
$$ r = \sqrt {(x^2 + y^2)}$$
$$ \theta = \tan({y \over x}) $$
Furthermore, the definitions of trigonometry tell us that These four expressions relating the coordinates (x, y) to the coordinates (r, $\theta$) apply only when $\theta$ is defined, as shown in Figure 3.

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