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Section 12.13 : Spherical Coordinates

3. Convert the Cylindrical coordinates for \(\left( {2,0.345, - 3} \right)\) into Spherical coordinates.

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From the point we’re given we have,

\[r = 2\hspace{0.5in}\theta = 0.345\hspace{0.5in}z = - 3\]

So, we already have the value of \(\theta \) for the Spherical coordinates.

Show Step 2

Next, we can determine \(\rho \).

\[\rho = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 3} \right)}^2}} = \sqrt {13} \] Show Step 3

Finally, we can determine \(\varphi \).

\[\cos \varphi = \frac{z}{\rho } = \frac{{ - 3}}{{\sqrt {13} }}\hspace{0.5in}\varphi = {\cos ^{ - 1}}\left( {\frac{{ - 3}}{{\sqrt {13} }}} \right) = 2.5536\]

The Spherical coordinates are then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\sqrt {13} ,0.345,2.5536} \right)}}\]