Polar Coordinates

You’ll be able to

Convert a point from polar form to rectangular form
Convert a point from rectangular form to polar form
Interpret r and θ as a distance and a direction

Locating a point by distance and angle

Instead of horizontal and vertical steps, polar coordinates locate a point by a distance and a direction: (r, θ), where r is how far the point is from the origin (the pole) and θ is the angle from the positive x-axis. The same point can be named many ways — adding 2π to θ, or using a negative r, lands you in the same place.

Converting between the two systems

Polar and rectangular coordinates are linked by right-triangle trigonometry. Going polar → rectangular, drop the point onto the axes: x = r cos θ and y = r sin θ. Going rectangular → polar, use the Pythagorean theorem for the distance, r = √(x² + y²), and the tangent ratio for the angle, θ = arctan(y/x) (adjusted for the correct quadrant).

Coordinate conversions

x = r cos θ · y = r sin θ · r = √(x² + y²) · tan θ = y/x

The first pair converts polar to rectangular; the second pair converts rectangular to polar. Always check the quadrant when finding θ.

Worked example

Convert the polar point (r, θ) = (2, π/2) to rectangular coordinates.

1.Use x = r cos θ: x = 2·cos(π/2) = 2·0 = 0.
2.Use y = r sin θ: y = 2·sin(π/2) = 2·1 = 2.
3.Combine the results into the rectangular point (0, 2).

Answer: The rectangular coordinates are (0, 2). At θ = π/2 the point lies straight up from the origin, a distance 2 along the y-axis.

Checkpoint

Convert the polar point (r, θ) = (2, π/2) to rectangular coordinates.

Watch out

Keep the roles straight: x uses cosine and y uses sine. Swapping them, x = r sin θ, sends the point to the wrong axis. Cosine is the horizontal (x) piece.

Checkpoint

For a rectangular point (x, y), the polar distance r is given by:

On the exam

When converting rectangular → polar, arctan alone can give the wrong quadrant, since it only returns angles in (−π/2, π/2). Sketch the point first and add π when it lies in the second or third quadrant.

Answer the 2 checkpoints as you read.