Question

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

Alternate coordinate systems
ANSWERED
asked 2021-08-02
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.

Expert Answers (1)

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