INTRODUCTION:

In honor of Descartes, The system used for describing the location of a point in a plane is called as the Cartesian system. In polar co-ordinate systems, the distance is used to determine the points from a fixed point and the direction is used to determine the angle. Polar co-ordinate system in the form of three dimensions is known as cylindrical co-ordinate system. We can change the polar form from Cartesian form using the derived formula.

Steps to solve Cartesian to polar:

Cartesian co-ordinates are denoted as (x,y) and polar co-ordinates are as (r, ?) . When we know the Cartesian co-ordinates then want to convert it into polar co-ordinates we need to solve a triangle using Pythagoras theorem,

r = sqrt (x2 + y2) { `sqrt(x^2+y^2)` }

Where,

r = Distance from origin to the point

x = Cartesian x co-ordinate

y = Cartesian y co-ordinate

We can find ? using tangent function,

? = tan -1 (y/x) {tan ? = y/ x}

Where,

T= angle relative to the zero axis.

Example problems to change cartesian coordinates to polar coordinates:

Example 1:

What is (4, 6) in polar co- ordinates?

Solution:

X= 4, Y= 6

Step 1: Find distance r

r = `sqrt(x^2+y^2)`

= `sqrt(4^2+6^2)`

=`sqrt(52)`

r = 7.211

Step 2: Find angle

?= tan -1 (y/x)

= tan-1(6/4)

= tan-1(1.5)

= 56.30°

Hence the polar form of (4,6) is (7.211, 56.30° )

Example 2:

Convert (-3,-3) into polar coordinates.

Solution:

X= -3, Y=-3

Step 1: Find distance r

r = `sqrt(x^2+y^2)`

= `sqrt(-3^2+ -3^2)`

= `sqrt(9+9)`

=`sqrt(18)`

r= 4.242

Step 2: Find angle

?= tan -1 (y/x)

= tan-1 ( -3/-3)

= tan-1(1)

= 45°

?= 225° {180°+45°= 225° since (-3,-3) lies on negative axis}

Hence the polar form of (-3,-3) is (4.242, 225° ).

Example 3:

Change (16, 12) to polar form.

Solution:

X=16, Y=12

Step 1: Find distance r

r = `sqrt(x^2+y^2)`

= `sqrt(16^2+12^2)`

=`sqrt(256+144)`

= `sqrt(400)`

r = 20

Step 2: Find angle

?= tan -1 (y/x)

= tan-1(12/16)

= tan-1(0.75)

= 36.87°

Hence the polar form of (16, 12) is (20, 36.87°).

Practice problems:

1.Find the polar form of (15, 7).

Answer: (16.6, 25?)

2.What is (14, 8) in polar co ordinates?

Answer: (16.12, 29.74 °).

In honor of Descartes, The system used for describing the location of a point in a plane is called as the Cartesian system. In polar co-ordinate systems, the distance is used to determine the points from a fixed point and the direction is used to determine the angle. Polar co-ordinate system in the form of three dimensions is known as cylindrical co-ordinate system. We can change the polar form from Cartesian form using the derived formula.

Steps to solve Cartesian to polar:

Cartesian co-ordinates are denoted as (x,y) and polar co-ordinates are as (r, ?) . When we know the Cartesian co-ordinates then want to convert it into polar co-ordinates we need to solve a triangle using Pythagoras theorem,