Given: spherical coordinates: \(\displaystyle{\left({8},{\frac{{\pi}}{{{3}}}},{\frac{{\pi}}{{{6}}}}\right)}\).

To find: convert into, cylindrical coordinates.

Solution: The spherical coordinates are given by \(\displaystyle{\left({s},\phi,\psi\right)}\)

So here \(\displaystyle{s}={8}.\phi={\frac{{\pi}}{{{3}}}}.\psi={\frac{{\pi}}{{{6}}}}\)

These coordinates can be converted into cylindrical coordinates by:

\(\displaystyle\gamma={s}{\sin{\psi}}\)

\(\displaystyle\phi=\phi\)

\(\displaystyle{z}={s}{\cos{\psi}}\)

So. \(\displaystyle\gamma={8}{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}.\phi={\frac{{\pi}}{{{3}}}}.{z}={8}{\cos{{\left({\frac{{\pi}}{{{6}}}}\right)}}}\).

Thus \(\displaystyle\gamma={8}\cdot{\frac{{{1}}}{{{2}}}}.\phi={\frac{{\pi}}{{{3}}}}.{z}={8}{\frac{{\sqrt{{{3}}}}}{{{2}}}}\)

Thus. \(\displaystyle\gamma={4}.\phi={\frac{{\pi}}{{{3}}}}.{z}={4}\sqrt{{{3}}}\)

To find: convert into, cylindrical coordinates.

Solution: The spherical coordinates are given by \(\displaystyle{\left({s},\phi,\psi\right)}\)

So here \(\displaystyle{s}={8}.\phi={\frac{{\pi}}{{{3}}}}.\psi={\frac{{\pi}}{{{6}}}}\)

These coordinates can be converted into cylindrical coordinates by:

\(\displaystyle\gamma={s}{\sin{\psi}}\)

\(\displaystyle\phi=\phi\)

\(\displaystyle{z}={s}{\cos{\psi}}\)

So. \(\displaystyle\gamma={8}{\sin{{\left({\frac{{\pi}}{{{6}}}}\right)}}}.\phi={\frac{{\pi}}{{{3}}}}.{z}={8}{\cos{{\left({\frac{{\pi}}{{{6}}}}\right)}}}\).

Thus \(\displaystyle\gamma={8}\cdot{\frac{{{1}}}{{{2}}}}.\phi={\frac{{\pi}}{{{3}}}}.{z}={8}{\frac{{\sqrt{{{3}}}}}{{{2}}}}\)

Thus. \(\displaystyle\gamma={4}.\phi={\frac{{\pi}}{{{3}}}}.{z}={4}\sqrt{{{3}}}\)