Spherical Coordinates — Question 2

Question 2

Convert the spherical point $(\rho,\theta,\phi)=(6,3\pi/4,\pi/3)$ to rectangular and cylindrical coordinates. Keep all values exact.

Question 2 – Solution

Strategy First compute the cylindrical radius $r=\rho\sin\phi$ and height $z=\rho\cos\phi$.

Cylindrical form $$r=6\sin\frac{\pi}{3}=3\sqrt 3,\qquad z=6\cos\frac{\pi}{3}=3.$$ Hence $\boxed{(r,\theta,z)=(3\sqrt 3,3\pi/4,3)}$.

Rectangular form $$x=3\sqrt 3\cos\frac{3\pi}{4}=-\frac{3\sqrt 6}{2},\qquad y=3\sqrt 3\sin\frac{3\pi}{4}=\frac{3\sqrt 6}{2}.$$ Therefore $$\boxed{(x,y,z)=\left(-\frac{3\sqrt 6}{2},\frac{3\sqrt 6}{2},3\right)}.$$