Spherical Coordinates — Question 2

Question 2 – Solution

Strategy. Use $r=\rho\sin\phi$ and $z=\rho\cos\phi$, then resolve $r$ using $\theta$.

Step 1: Cylindrical data $$r=8\sin(\pi/3)=4\sqrt 3,\qquad z=8\cos(\pi/3)=4.$$ Thus the principal cylindrical coordinates are $$\boxed{(r,\theta,z)=(4\sqrt 3,5\pi/6,4)}.$$

Step 2: Cartesian data $$x=r\cos\theta=(4\sqrt 3)(-\sqrt 3/2)=-6,\qquad y=r\sin\theta=(4\sqrt 3)(1/2)=2\sqrt 3.$$ Therefore $$\boxed{(x,y,z)=(-6,2\sqrt 3,4)}.$$

Verification. $x^2+y^2=36+12=48=r^2$, and $x^2+y^2+z^2=48+16=64=\rho^2$.