Question

How do you convert rectangular coordinates to polar coordinates?

Answer 1

The students should know the importance of both the types of coordinates, where they are used and why we convert them from one form to another. A rectangular coordinate system is a set of $2$ real numbers that intersect at right angles. Polar coordinate system is a $2$ dimensional coordinate system in which each point in the plane is determined by a distance from a reference point and an angle from the reference direction. When the student has to find the value of radius for a curve like circle or parabola, he/she has to use polar coordinates.

Complete step by step solution:
Let us consider the coordinates in the rectangular system as $(x,y)$
Co-ordinates in the polar form would be $(r,\theta )$.
Now in order to convert polar form the student has to equate the coordinates in polar form as follows
$x = r\cos \theta .......(1)$
$y = r\sin \theta ........(2)$
Also if the student wants to find the value of $r$, we square the above equations and add them .
${r^2} = {x^2} + {y^2}$
To find the angle, divide $2$ by 1$$
$\tan \theta = \dfrac{y}{x}$

In order to convert rectangular coordinates to polar coordinates, the student has to use $x = r\cos \theta$ & $y = r\sin \theta$.

Note:
The conversion is extremely important when the angle between the curves is asked or when the radius of the curve is to be found out. A student may apply the formula to a set of coordinates given , For example a student may be given rectangular coordinates $(2,2\sqrt 3 )$ and the question is to convert to polar coordinates. In this sum the first step would be to first find out the value of $r$ by using ${r^2} = {x^2} + {y^2}$, and then $\theta$ by using $\tan \theta = \dfrac{y}{x}$. For all the sums, students have to follow the same procedure.