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Polar Coordinates

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In mathematics, the coordinate system plays a very important role, especially in geometry. It is a system of numbers, known as coordinates, which are utilized to uniquely position a point or a geometric element on the Euclidean space. Every object is positioned in space. The coordinate system is way to determine this position. The coordinates may have one or more tuples, i.e. they have a set of values representing an exact position of something. The order of these values is quite important. On the graphs and maps, you would have commonly seen a pair of numbers showing where a point is located. Here, the first number would indicate distance left or right and the second number denotes the distance up or down.

Mainly, there are following types of coordinate systems:

1) Number line

2) Cartesian coordinate system

3) Cylindrical coordinate system

4) Polar coordinate system

5) Spherical coordinate system


Here, we are going to discuss about polar coordinate system and its properties.

System

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The polar coordinate system is a type of coordinate system that is usually two-dimensional. In this coordinate system, each point on a plane is to be specified by the distance from a certain point, called the reference point as well as by an angle from a certain direction, called the reference direction. In polar coordinate system, the reference point is better known as the pole and the reference direction is said to be the polar axis.

Have a look at the following diagram of this system:
Polar Coordinate System

The polar coordinate is made of two elements: one is radial coordinate, while another is known as angular coordinate. The radial coordinate refers to the distance of point from the pole. This coordinate is simply termed as the radius. Whereas, the angular coordinate denotes the angle of given point from polar axis. This coordinate is also called polar angle or azimuth.

Conversion

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Let r and $\theta$ be the radial and angular coordinates in the polar coordinate system, then they can be converted into Cartesian coordinates using the following relations :
r cos $\theta$ and r sin $\theta$
where, $\theta$ is said to be the counterclockwise angle from the x-axis in Cartesian coordinate system. The values of r and $\theta$ can be calculated as below.

Squaring and adding above two equations, we get

$x^{2} + y^{2} = r^{2} cos^{2} \theta + r^{2} sin^{2} \theta$

$x^{2} + y^{2} = r^{2} (cos^{2} \theta + r^{2} sin^{2} \theta)$

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

Dividing second equation from first, we obtain

$\frac{y}{x}$ = $\frac{sin \theta}{cos \theta}$

$\frac{y}{x}$ = $tan \theta$
$\theta$ = $tan^{-1}$ $\frac{y}{x}$

Cylindrical

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When the polar coordinates are defined with respect to a cylinder, they are known as the cylindrical polar coordinates. Such coordinates are actually three dimensional coordinates. 

The diagram of cylindrical polar coordinate system is shown below:

Cylindrical Polar Coordinate
In this type of coordinate system, the axis of a right circular cylinder is considered as z-axis. From this axis, the perpendicular distance from the cylinder axis is denoted by radial coordinate and is denoted by r. Also, the angle of radius from horizontal axis is known as the azimuthal angle and is represented by symbol $\phi$. We denoted the cylindrical polar coordinates in the form of (r, $\phi$, z).

Spherical

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The polar coordinates can also be denoted with respect to a sphere. In this case, they are called the spherical polar coordinates. These coordinates are defined in three dimensional system. 

Let us have a look at the diagram of spherical polar coordinate system below:

Spherical Polar Coordinate
This coordinate system considers center of sphere as the origin of coordinate system. Radial distance is denoted by r. The angle of radial distance with vertical axis is considered as angular coordinate and is represented by $\theta$, while of radial distance angle from horizontal axis is called azimuthal angle and is denoted by $\phi$. The spherical polar coordinates has three tuples of the form (r, $\theta$, $\phi$).

Grapher

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In order to graph polar coordinates, we need to follow the steps written below:

Let us suppose the given polar coordinates are (r, $\theta$):

Step 1: Take a point (say O) as the center.

Step 2: From this point, a horizontal line (radial axis) is drawn.

Step 3: From this radial axis, the given azimuthal angle $\theta$ is inclined in the counterclockwise direction. If angle is negative, it should be inclined in clockwise direction.

Step 4: At this line, we denote given radial coordinate r. If r is negative, it should be shown in the reverse direction of this line.


Have a look at the following examples.

i) (1, $\frac{5 \pi}{4}$)
Polar Coordinate Example

ii) (2, 3$\pi$)
Example of Polar Coordinate

iii) (1, $\frac{13 \pi}{4}$)
Polar Coordinate Problem

ii) (1, $\frac{-3 \pi}{4}$)
Problem of Polar Coordinate