Question

Change the following from cylindrical coordinates to rectangular coordinates
1) (5, π/6, 3)
2) (6, π/3, -5)

Answer 1

Solution: We know that "The cylindrical coordinates are denoted by $(r,\theta,z)$ and rectangular coordinates are denoted by $(x,y,z)$ "

To convert from cylindrical coordinates to rectangular coordinates we use the equations

$x=r~cos~\theta$

$y=r~sin~\theta$ and

$z=z$

(1) Given cylindrical coordinates are $(5, \frac{\pi}{6},3)$

$\therefore (r,\theta,z)=(5, \frac{\pi}{6},3)$

Now, to find rectangular coordinates, we have

$x=r~cos~\theta=5~cos~(\frac{\pi}{6})=5(\frac{\sqrt{3}}{2})=\frac{5\sqrt{3}}{2}$

$y=r~sin~\theta=5~sin~(\frac{\pi}{6})=5(\frac{1}{2})=\frac{5}{2}$

$z=z=3$

Therefore rectangular coordinates $(x,y,z)=(\frac{5\sqrt{3}}{2},\frac{5}{2},3)$

(2) Given cylindrical coordinates are $(6, \frac{\pi}{3},-5)$

$\therefore (r,\theta,z)=(6, \frac{\pi}{3},-5)$

Now, to find rectangular coordinates, we have

$x=r~cos~\theta=6~cos~(\frac{\pi}{3})=6(\frac{1}{2})=\frac{6}{2}=3$

$y=r~sin~\theta=6~sin~(\frac{\pi}{3})=6(\frac{\sqrt{3}}{2})=\frac{6\sqrt{3}}{2}=3\sqrt{3}$

$z=z=-5$

Therefore rectangular coordinates $(x,y,z)=(3,3\sqrt{3},-5)$