Question

# Convert between the coordinate systems spherical: (8, \frac{\pi}{3}, \frac{\pi}{6}) Change to cylindrical.

Alternate coordinate systems
Convert between the coordinate systems. Use the conversion formulas and show work.
spherical: $$\displaystyle{\left({8},{\frac{{\pi}}{{{3}}}},{\frac{{\pi}}{{{6}}}}\right)}$$
Change to cylindrical.

2021-08-03
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. $$\displaystyle\gamma={4}.\phi={\frac{{\pi}}{{{3}}}}.{z}={4}\sqrt{{{3}}}$$