From polar to Cartesian

Statement

Convert to Cartesian coordinates: (a) $(\rho,\theta) = (3,\pi/6)$; (b) $(2,2\pi/3)$; (c) $(\sqrt 2,\,-\pi/4)$.

Solution

Use $x = \rho\cos\theta$, $y = \rho\sin\theta$.

• (a) $x = 3\cos\tfrac{\pi}{6} = 3\cdot\tfrac{\sqrt3}{2} = \tfrac{3\sqrt3}{2}$, $\ y = 3\sin\tfrac{\pi}{6} = \tfrac{3}{2}$: $\boxed{\left(\tfrac{3\sqrt3}{2},\,\tfrac{3}{2}\right)}$.
• (b) $x = 2\cos\tfrac{2\pi}{3} = 2\cdot\left(-\tfrac12\right) = -1$, $\ y = 2\sin\tfrac{2\pi}{3} = 2\cdot\tfrac{\sqrt3}{2} = \sqrt3$: $\boxed{(-1,\,\sqrt3)}$.
• (c) $x = \sqrt2\cos\left(-\tfrac{\pi}{4}\right) = \sqrt2\cdot\tfrac{\sqrt2}{2} = 1$, $\ y = \sqrt2\sin\left(-\tfrac{\pi}{4}\right) = -1$: $\boxed{(1,\,-1)}$.

Links

Topics: Goniometry
Concepts: Cartesian coordinates · Polar coordinates · Cartesian plane
Methods: Polar–Cartesian conversion
Skills: Calculating · Using formulae
Exercise types: Evaluating expressions