The polar coordinates can be easily identified with the visualisation of an analogue clock. A small piece of mental task can be identified for the proper conceptualization of polar coordinates. If for example it is imagined that the time that is displayed in a clock is 3:30, defer the hour hand would be pointing at 3 and the minute hand be pointing at 6. If it is asked that the description of the location of the hour hand need to be provided with respect to the location of the minute hand, they might be some exact measurements taken for each of the numbers but it can be represented in the most concise way of answering by saying that the hour hand and the minute hand pointing at 3 and 6 respectively are 90 degrees apart. Therefore this example describes how, in a nutshell, the polar coordinates can be simplified by locating the points in a graph and represent the exact location of the coordinates with the help of the distance between them.
The formal definition of the polar coordinates can be defined by the set of values that specify the location of any said point relating to the fact of the distance that is existing between the point and the fixed origin and also the angle between the point and a particular fixed direction. The polar coordinates are used to build up their complementary system for Cartesian coordinates. This Cartesian co-ordinate is a location moving across the x-axis towards the y-axis in a rectangular fashion represented as (x, y) where is the polar coordinates are represented as (r, θ).
The polar coordinates and Cartesian coordinates can also be converted into each other, where the following section mood describe in details how to convert Cartesian coordinates to polar coordinates:
Polar coordinates and the Cartesian coordinates are the most commonly used systems and therefore a set of Cartesian coordinates can also be converted into a setup polar coordinates.
For example if the coordinates (x, y) = (3, 4), then it can be converted into polar coordinates by creating a triangle with base on the x-axis and the vertex taken as (3, 4).
The shortest distance that can be observed between the two points would be the point taken and the origin. After linking these two points the hypotenuse of the triangle, that is the longest side of the triangle, can be obtained. Therefore this would be the first point of the polar coordinates where r would be represented as (r, θ). With the help of the Pythagorean Theorem, the value of (r, θ) can be obtained.
Therefore, deriving the Pythagoras theorem to find the value of θ, it can be clearly said that the determination of the angle between the hypotenuse and the fixed direction would be calculated to three possible inverse trigonometric expressions, and the value of θ can thus be obtained. The Cartesian coordinates taken in this example are (3, 4) to which it can be said that the polar coordinates should be (5, 53.1 3°).
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