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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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Start Solution

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,

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