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Section 12.13 : Spherical Coordinates
2. Convert the Cartesian coordinates for \(\left( { - 2, - 1, - 7} \right)\) into Spherical coordinates.
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Start SolutionFrom the point we’re given we have,
\[x = - 2\hspace{0.5in}y = - 1\hspace{0.5in}z = - 7\] Show Step 2Let’s first determine \(\rho \).
\[\rho = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( { - 1} \right)}^2} + {{\left( { - 7} \right)}^2}} = \sqrt {54} \] Show Step 3We can now determine \(\varphi \).
\[\cos \varphi = \frac{z}{\rho } = \frac{{ - 7}}{{\sqrt {54} }}\hspace{0.5in}\varphi = {\cos ^{ - 1}}\left( {\frac{{ - 7}}{{\sqrt {54} }}} \right) = 2.8324\] Show Step 4Let’s use the \(y\) conversion formula to determine \(\theta \).
\[\sin \theta = \frac{{ - 1}}{{\rho \sin \varphi }} = \frac{{ - 1}}{{\sqrt {54} \sin \left( {2.8324} \right)}} = - 0.4472\hspace{0.25in} \to \hspace{0.25in}{\theta _1} = {\sin ^{ - 1}}\left( { - 0.4472} \right) = - 0.4636\]This angle is in the fourth quadrant and if we sketch a quick unit circle we see that a second angle in the third quadrant is \({\theta _2} = \pi + 0.4636 = 3.6052\).
If we look at the three dimensional coordinate system from above we can see that from our \(x\) and \(y\) coordinates the point is in the third quadrant. This in turn means that we need to use \({\theta _2}\) for our point.
The Spherical coordinates are then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\sqrt {54} ,3.6052,2.8324} \right)}}\]