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:

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

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

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

Question	Answer
How do you convert a polar coordinate (r, \theta) to cartesian (x, y)?	Use the identities x = r\cos\theta and y = r\sin\theta, then simplify to the form (x, y).
What are the four polar coordinate identities?	x = r\cos\theta, y = r\sin\theta, r^2 = x^2 + y^2, \tan\theta = \frac{y}{x}
What are the components of a polar coordinate (r, \theta)?	r is the magnitude (distance from the origin), \theta is the bearing from the right-stretching line going anticlockwise.
What are the components of a cartesian coordinate (x, y)?	x is the horizontal distance from the origin, y is the vertical distance from the origin.
What are the steps to convert polar to cartesian coordinates?	Substitute r and \theta into x = r\cos\theta and y = r\sin\theta, then simplify to get (x, y).
Convert (4\sqrt2, \frac{\pi}{4}) to cartesian.	(4, 4) because x = 4\sqrt2 \cdot \frac{\sqrt2}{2} = 4 and y = 4\sqrt2 \cdot \frac{\sqrt2}{2} = 4.
Convert (4.123, 1.816) to cartesian to 3sf.	(-1.00, 4.00) because x = 4.123\cos1.816 \approx -1.00 and y = 4.123\sin1.816 \approx 4.00.
Convert (5, 2.356) to cartesian to 3sf.	(-3.54, 3.54) because x = 5\cos2.356 \approx -3.54 and y = 5\sin2.356 \approx 3.54.
Convert (3, 4.712) to cartesian to 3sf.	(0.00, -3.00) because x = 3\cos4.712 \approx 0.00 and y = 3\sin4.712 \approx -3.00.
Convert (2, 5.890) to cartesian to 3sf.	(1.73, -1.00) because x = 2\cos5.890 \approx 1.73 and y = 2\sin5.890 \approx -1.00.