The Lesson

Polar coordinates can be converted to Cartesian coordinates using the following formulas: In these formulas:
The graph below shows what we mean by the same point defined in polar coordinates (r, θ) and Cartesian coordinates (x, y):

How to Convert from Polar to Cartesian Coordinates

Converting from the polar to the Cartesian coordinates of a point is easy.

Question

What is a point described by the polar coordinates (8, 30°) in Cartesian coordinates?

1

x-coordinate = r cos θ
Note: cos θ is the cosine of the angle.

2

Find r and θ from the polar coordinates given in the question. In our example, the polar coordinates of the point is (8, 30°). They are represented in the formula by (r, θ).
(r, θ) = (8, 30°) ∴ r = 8, θ = 30°

3

Substitute r and θ into the formula.

x-coordinate = 8 cos (30°)

x-coordinate = 8 × 0.87

x-coordinate = 6.9

The x-coordinate is 6.9

4

y-coordinate = r sin θ
Note: sin θ is the sine of the angle.

5

Substitute r and θ into the formula.

y-coordinate = 8 sin (30°)

y-coordinate = 8 × 0.5

y-coordinate = 4

The y-coordinate is 4

6

Write down the Cartesian coordinates as a pair of numbers in brackets, separated by a comma. The x-coordinate (6.9) found in Step 3 goes on the left. The y-coordinate (4) found in Step 5 goes on the right.

The polar coordinates (8, 30°) become (6.9, 4) when converted to Cartesian coordinates.

Lesson Slides

The slider below gives another example of how to convert from polar to Cartesian coordinates.