Spherical Coordinates — Question 1

Question 1

Convert the point $(1,\sqrt 3,2\sqrt 3)$ to spherical coordinates, using $\rho\ge 0$, $0\le\theta<2\pi$, and $0\le\phi\le\pi$. Verify the result.

Question 1 – Solution

Strategy Find distance, azimuth, and polar angle in that order.

Coordinates $$\rho=\sqrt{1+3+12}=4,\qquad \theta=\arctan(\sqrt 3)=\frac{\pi}{3}.$$ Also $\cos\phi=z/\rho=\sqrt 3/2$, so $\phi=\pi/6$. Thus $$\boxed{(\rho,\theta,\phi)=\left(4,\frac{\pi}{3},\frac{\pi}{6}\right)}.$$

Verification Using $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, and $z=\rho\cos\phi$ gives $1,\sqrt 3,2\sqrt 3$, respectively.