Polar to Cartesian coordinates

Let’s suppose we want to convert a coordinate in the form to a coordinate in the form . How do we do that?

Again, we’ll be using our polar coordinate identities a lot to do this. Here they are summarised:

Form of coordinates

• Polar coordinates are in the form , where:
  • is the magnitude, the distance of a point from the origin
  • is the bearing from the origin, starting from the right-stretching line and going anticlockwise.
• Cartesian coordinates are in the form , where:
  • is the horizontal distance of a point from the origin
  • is the vertical distance of a point from the origin

General steps

To convert polar to cartesian, we can just use the first two identities:

Substitute in the values for and , and then simplify to get the answer as a cartesian coordinate in the form .

Examples

Convert the polar coordinate to cartesian coordinates

• Answer:

Convert the polar coordinate to cartesian coordinates (to 3sf)

• Answer:

Convert the polar coordinate to cartesian coordinates (to 3sf)

• Answer:

Convert the polar coordinate to cartesian coordinates (to 3sf)

• Answer:

Convert the polar coordinate to cartesian coordinates (to 3sf)