Conversion between polar and Cartesian

The two descriptions of a point, polar and Cartesian, are converted into each other using sine, cosine and the arctangent.

Property — Conversion polar ↔ Cartesian

\text{(polar} \to \text{Cartesian)}\quad & x = \rho\cos\theta, & y = \rho\sin\theta. \\ \text{(Cartesian} \to \text{polar)}\quad & \rho = \sqrt{x^2+y^2}, & \theta = \arctan\!\left(\frac{y}{x}\right)\ \text{(with a choice of quadrant)}. \end{aligned}$$

Warning — The quadrant in the inversion

$\arctan(y/x)$ returns an angle in $(-\pi/2, \pi/2)$: it must be corrected by looking at the signs of $x$ and $y$. In many languages (Python, C, MATLAB) there is the function atan2$(y,x)$ which does the right thing, returning $\theta\in(-\pi,\pi]$.

Example — Conversion both ways

$P(x,y) = (1,\sqrt 3)$ in polar: $\rho = \sqrt{1+3} = 2$, $\theta = \arctan(\sqrt 3) = \pi/3$ (first quadrant). Polar coordinates: $(2,\pi/3)$.

$Q(\rho,\theta) = (4,\,3\pi/4)$ in Cartesian: $x = 4\cos(3\pi/4) = -2\sqrt 2$, $y = 4\sin(3\pi/4) = 2\sqrt 2$. So $Q = (-2\sqrt 2,\,2\sqrt 2)$.

Links

Topics: Trigonometry
Concepts: Arctangent · Cartesian coordinates · Polar coordinates · Cartesian plane
Methods: Polar-Cartesian conversion
Skills: Calculating · Reasoning by cases · Using formulae