The LessonPolar coordinates can be converted to Cartesian coordinates using the following formulas: In these formulas:
- x is the x-coordinate of the point in Cartesian coordinates.
- y is the y-coordinate of the point in Cartesian coordinates.
- r is the radial coordinate of the point in polar coordinates.
- θ is the angular coordinate of the point in polar coordinates.
How to Convert from Polar to Cartesian CoordinatesConverting from the polar to the Cartesian coordinates of a point is easy.
QuestionWhat is a point described by the polar coordinates (8, 30°) in Cartesian coordinates?
Find the X-Coordinate
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°
Substitute r and θ into the formula.The x-coordinate is 6.9
x-coordinate = 8 cos (30°)
x-coordinate = 8 × 0.87
x-coordinate = 6.9
Find the Y-Coordinate
Start with the formula:
y-coordinate = r sin θNote: sin θ is the sine of the angle.
Substitute r and θ into the formula.The y-coordinate is 4
y-coordinate = 8 sin (30°)
y-coordinate = 8 × 0.5
y-coordinate = 4
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.