Answer
a) $( 0, 2, 0)$
b) $(\sqrt 6,-\sqrt 6,2)$
Work Step by Step
$(\rho,\theta,\phi)\rightarrow(x,y,z)$
$x=\rho\sin\phi\cos\theta$
$y=\rho\sin\phi\sin\theta$
$z=\rho\cos\phi$
a) $(\rho,\theta,\phi)=\left(2,\frac{\pi}{2},\frac{\pi}{2}\right)$
$x=2\sin \frac{\pi}{2}\cos\frac{\pi}{2}=2(1)(0)=0$
$y=2\sin\frac{\pi}{2}\sin\frac{\pi}{2}=2(1)(1)=2$
$z=2\cos\frac{\pi}{2}=2(0)=0$
$(x,y,z)=(0,2,0)$
b) $(\rho,\theta,\phi)=\left(4,-\frac{\pi}{4},\frac{\pi}{3}\right)$
$x=4\sin \frac{\pi}{3}\cos\left(-\frac{\pi}{4}\right)=4\left(\frac{\sqrt 3}{2}\right)\left(\frac{\sqrt 2}{2}\right)=\sqrt 6$
$y=4\sin\frac{\pi}{3}\sin\left(-\frac{\pi}{4}\right)=4\left(\frac{\sqrt 3}{2}\right)\left(-\frac{\sqrt 2}{2}\right)=-\sqrt 6$
$z=4\cos\frac{\pi}{3}=4\left(\frac{1}{2}\right)=2$
$(x,y,z)=(\sqrt 6,-\sqrt 6,2)$
