Vivekanand Vellanki
3

You have the same polar co-ordinates for (3, 4) and (-4, 3) - this is incorrect. For (-4, 3), the angle would be more than 90. For (-4, 3), the polar co-ordinates would be (5, 90+53).

The purpose of any co-ordinate system is to be able to describe a point on a plane precisely.

Imagine you had a laser that was pointed East. This light from the laser covers every point on the x-axis, where x is positive. So, to describe any point on the x-axis, all we need to know is the distance from the origin and the direction in which the laser is pointed.

Now, rotate the laser slowly. Once an entire 360 degree rotation is complete, the light would have covered all the points on the plane.

This indicates that all you need to describe a point in a plane is the distance of the point from the origin (r, in polar co-ordinates) and the direction in which the laser is pointed (\theta, in polar co-ordinates).

A point (a, b) in the cartesian system is at a distance of a units from the origin towards the x-axis and at a distance of b units from the origin towards the y-axis.

The distance of this point from the origin is: r=\sqrt{a^2+b^2}. The direction of the laser would such that \tan\theta=\frac{b}{a}; or \theta=\tan^{-1}\left(\frac{b}{a}\right)

Hence, (a, b) in the cartesian system equals \left(\sqrt{a^2+b^2},\ \tan^{-1}\frac{b}{a}\right)in polar co-ordinates.

Mahesh Godavarti
2

The most important thing to remember about co-ordinates is that they help you locate an object on a surface or in space. Essentially, they are giving directions to the object from a reference point called the origin.

The co-ordinates allow you to answer the question - Where is the object located? Or How do I get to the object?

They are many ways to answer the question. I will describe two.

1. The first way is: From the origin walk x steps East and then y steps North. You will arrive at the object. This would be the rectangular coordinate system.
2. The second way is: While standing on the origin, start off by facing East. Then turn \theta \degree counter-clockwise. Then walk r steps in the direction you are facing. You will arrive at the object. This would be the polar coordinate system.

Note that knowing one set of directions to arrive at the object, allows you to come up with the second set of directions as well.

Let's take the following example:

Let's describe how to arrive at the black dot on the 2D plane. If we use the rectangular coordinates (or Cartesian coordinates) to describe how to arrive at the black dot, we would say from the origin, walk 3 steps East and then walk 4 steps North (or walk 4 steps North and then walk 3 steps East).

Alternatively, we can use the polar coordinates to describe how to arrive at the dot. We can say start off by facing East, turn \theta \degree = \tan^{-1} \frac{4}{3} counterclockwise and then walk \sqrt{3^2 + 4^2} = 5 steps in the direction you are facing.

Let's use this to figure out how to come up with polar coordinates for the examples you have on your page.

Rectangular (Cartesian) coordinates (2, 0), then the polar coordinates would be (2, 0 \degree) . Do you see how?