Polar to Cartesian Coordinates

A point in the plane can be described in two ways. Cartesian coordinates give a horizontal and a vertical distance, $(x,\ y)$ . Polar coordinates give a distance from the origin and an angle, $(r,\ \theta)$ . Both name the same point, so there must be a rule that carries one into the other.

Concept The two coordinate systems

x = r\cos\theta \qquad y = r\sin\theta

r — the distance from the origin to the point.

θ — the angle measured from the positive x-axis.

Dropping a perpendicular from the point makes a right triangle whose legs are exactly x and y.

This is the unit circle with the radius released from 1. There, $(\cos\theta,\ \sin\theta)$ was the point itself; here every coordinate is scaled by $r$ .

Example Converting

(4,\ \pi/6)

Write the polar point $r = 4$ , $\theta = \pi/6$ in Cartesian form.

\pi/6 = 30°

x = r\cos\theta = 4\cos 30° = 4 \times \dfrac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.46

y = r\sin\theta = 4\sin 30° = 4 \times \dfrac{1}{2} = 2

Both coordinates are positive, so the point lies in the first quadrant.

⟹ (x, y) = (2√3, 2)

Note Negative r and negative angles

Neither a negative radius nor a negative angle needs a special rule. A negative $r$ points the other way along the same line — a turn of 180°. A negative angle is measured clockwise, and the parity of the two functions absorbs it:

\cos

is even:

\cos(-\theta) = \cos\theta

\sin

is odd:

\sin(-\theta) = -\sin\theta

Example Converting

(-6,\ -120°)

Here both the radius and the angle are negative. Apply the same two formulas and let the parity rules do the work.

x = (-6)\cos(-120°)

, and

\cos(-120°) = \cos 120° = -\tfrac{1}{2}

x = (-6)\left(-\tfrac{1}{2}\right) = 3

y = (-6)\sin(-120°)

, and

\sin(-120°) = -\sin 120° = -\dfrac{\sqrt{3}}{2}

y = (-6)\left(-\dfrac{\sqrt{3}}{2}\right) = 3\sqrt{3} \approx 5.20

Two negatives have cancelled twice, so the point lands in the first quadrant.

⟹ (x, y) = (3, 3√3)

Summary

Polar coordinates (r, θ) convert to Cartesian by x = r cos θ and y = r sin θ.
The rule is the unit circle with the radius no longer fixed at 1.
cos is an even function and sin is an odd function, which handles every negative angle.
A negative r reverses the direction of the point by 180°; it is not an error.