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Polar and Cartesian Coordinates

To pinpoint where you are on a map or graph there are two main systems:

Cartesian Coordinates

Using Cartesian Coordinates you mark a point by how far along and how far up it is:

Polar Coordinates

Using Polar Coordinates you mark a point by how far away, and what angle it is:

Converting

To convert from one to the other, you need to solve the triangle:


To Convert from Cartesian to Polar

If you have a point in Cartesian Coordinates (x,y) and need it in Polar Coordinates (r,θ), you need to solve a triangle where you know two sides.

Example: What is (12,5) in Polar Coordinates?

Use Pythagoras Theorem to find the long side (the hypotenuse):

r2 = 122 + 52

r = √ (122 + 52)

r = √ (144 + 25) = √ (169) = 13

Use the Tangent Function to find the angle:

tan( θ ) = 5 / 12

θ = atan( 5 / 12 ) = 22.6 °

So, to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

r = √ (x2 + y2)

θ = atan( y / x )

 

To Convert from Polar to Cartesian

If you have a point in Polar Coordinates (r, θ), and need it in Cartesian Coordinates (x,y) you need to solve a triangle where you know the long side and the angle:

Example: What is (13, 23 °) in Cartesian Coordinates?



Use the Cosine Function for x: cos( 23 °) = x / 13
Rearranging and solving: x = 13 × cos( 23 °) = 13 × 0.921 = 11.98
   
Use the Sine Function for y: sin( 23 °) = y / 13
Rearranging and solving: y = 13 × sin( 23 °) = 13 × 0.391 = 5.08

 

So, to convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) :

x = r × cos( θ )

y = r × sin( θ )

And that is it !