Question

Find the spherical coordinates of the point whose coordinates are
1) (1, 1, √6)
2) (-2, 2√3, 4)
3) (-√3, 1, -2)
4) (4,-4√3, 6)

Answer 1

We remind the expressions of spherical coordinates :

$\rho = \sqrt{x^2+y^2+z^2}$

$\theta = \arccos(\frac{z}{\rho})$

$\phi$ such that $x=\rho \sin(\theta) \cos(\phi), y=\rho\sin(\theta)\sin(\phi)$

Therefore we have :

1. $(\rho,\theta,\phi) = (2\sqrt{2}, \frac{\pi}{6}, \frac{\pi}{4})$
2. $(4\sqrt{2}, \frac{\pi}{4}, \frac{2\pi}{3})$
3. $(2\sqrt{2}, \frac{3\pi}{4}, \frac{5\pi}{6})$
4. $(10, \arccos(\frac{3}{5}), -\frac{\pi}{3})$