From Cartesian to polar

Statement

Convert to polar coordinates (with $\theta\in[0,2\pi)$): (a) $(3,0)$; (b) $(0,5)$; (c) $(-1,1)$; (d) $(-1,-\sqrt 3)$.

Solution

Use $\rho = \sqrt{x^2+y^2}$ and choose $\theta$ according to the quadrant.

• (a) $\rho = 3$, the point lies on the positive $x$-semiaxis: $\theta = 0$. $\boxed{(3,\,0)}$.
• (b) $\rho = 5$, the point lies on the positive $y$-semiaxis: $\theta = \tfrac{\pi}{2}$. $\boxed{\left(5,\,\tfrac{\pi}{2}\right)}$.
• (c) $\rho = \sqrt{1+1} = \sqrt2$; the point $(-1,1)$ is in the second quadrant: $\theta = \tfrac{3\pi}{4}$. $\boxed{\left(\sqrt2,\,\tfrac{3\pi}{4}\right)}$.
• (d) $\rho = \sqrt{1+3} = 2$; the point $(-1,-\sqrt3)$ is in the third quadrant, with reference angle $\arctan\tfrac{\sqrt3}{1} = \tfrac{\pi}{3}$: $\theta = \pi + \tfrac{\pi}{3} = \tfrac{4\pi}{3}$. $\boxed{\left(2,\,\tfrac{4\pi}{3}\right)}$.

Links

Topics: Goniometry
Concepts: Arctangent · Cartesian coordinates · Polar coordinates · Cartesian plane
Methods: Polar–Cartesian conversion
Skills: Reasoning by cases · Using formulae
Exercise types: Evaluating expressions