Answer
$(\frac{3\sqrt 3}{2}, \frac{3}{2}, -3\sqrt 3)$
Work Step by Step
We have the spherical coordinates:
$(\rho, \theta, \phi)=(6,\frac{\pi}{6}, \frac{5\pi}{6})$
$x= \rho \sin\phi \cos \theta=6\sin\frac{5\pi}{6}\cos \frac{\pi}{6}$
$=6\times\sin(\pi-\frac{\pi}{6})\times\frac{\sqrt 3}{2}$
$=6\times\sin \frac{\pi}{6}\times\frac{\sqrt 3}{2}$ ( As $\sin(\pi-x)=\sin x$)
$=6\times\frac{1}{2}\times\frac{\sqrt 3}{2}=\frac{3\sqrt 3}{2}$
$y=\rho\sin\phi\sin\theta=6\sin\frac{5\pi}{6}\cos \frac{\pi}{6}$
$=6\times\frac{1}{2}\times\frac{1}{2}=\frac{3}{2}$
$z=\rho\cos\phi= 6\times\cos \frac{5\pi}{6}$
$=6\times\cos(\pi-\frac{\pi}{6})$
$=6\times(-\cos \frac{\pi}{6})$ ( As $\cos (\pi-x)=-\cos x$)
$=6(-\frac{\sqrt 3}{2})=-3\sqrt 3$
Therefore, the rectangular coordinates are
$(x,y,z)=(\frac{3\sqrt 3}{2}, \frac{3}{2}, -3\sqrt 3)$
