Question

How to convert polar coordinates to rectangular coordinates?

Answer 1

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

coordinates cartesian (12,5)

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

coordinates polar 13 at 22.6 degrees

Converting
To convert from one to the other we will use this triangle:

coordinates triangle

To Convert from Cartesian to Polar
When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,θ) we solve a right triangle with two known sides.

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

coordinates to polar

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

r2 = 122 + 52
r = √ (122 + 52)
r = √ (144 + 25)
r = √ (169) = 13
Use the Tangent Function to find the angle:

tan( θ ) = 5 / 12
θ = tan-1 ( 5 / 12 ) = 22.6° (to one decimal)
Answer: the point (12,5) is (13, 22.6°) in Polar Coordinates.

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