Polar to Cartesian coordinates

Let’s suppose we want to convert a coordinate in the form (r,\theta) to a coordinate in the form (x,y). How do we do that?

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

• x=r\cos\theta
• y=r\sin\theta
• r^2=x^2+y^2
• \tan\theta=\frac yx

Form of coordinates

• polar coordinates are in the form (r,\theta), where:
  • r is the magnitude, the distance of a point from the origin
  • \theta is the bearing from the origin, starting from the right-stretching line and going anticlockwise.
• cartesian coordinates are in the form (x,y), where:
  • x is the horizontal distance of a point from the origin
  • y 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:

• x=r\cos\theta
• y=r\sin\theta

Substitute in the values for r and \theta, and then simplify to get the answer as a cartesian coordinate in the form (x,y).

Examples

Convert the polar coordinate (4\sqrt2, \frac{\pi}4) to cartesian coordinates

• x=4\sqrt2\cos\frac{\pi}4
• x=4\sqrt2\cdot\frac{\sqrt2}2
• x=4
• y=4\sqrt2\sin\frac{\pi}4
• y=4\sqrt2\cdot\frac{\sqrt2}2
• y=4
• Answer: (4,4)

Convert the polar coordinate (4.123, 1.816) to cartesian coordinates (to 3sf)

• x=4.12\cos1.816
• x=-1.00
• y=4.12\sin1.816
• y=4.00
• Answer: (-1.00, 4.00)

Convert the polar coordinate (5, 2.356) to cartesian coordinates (to 3sf)

• x=5\cos2.356
• x=-3.54
• y=5\sin2.356
• y=3.54
• Answer: (-3.54, 3.54)

Convert the polar coordinate (3, 4.712) to cartesian coordinates (to 3sf)

• x=3\cos4.712
• x=0.00
• y=3\sin4.712
• y=-3.00
• Answer: (0.00, -3.00)

Convert the polar coordinate (2, 5.890) to cartesian coordinates (to 3sf)

• x=2\cos5.890
• x=1.73
• y=2\sin5.890
• y=-1.00