Section 12.13 : Spherical Coordinates
6. Convert the equation written in Spherical coordinates into an equation in Cartesian coordinates.
\[\csc \varphi = 2\cos \theta + 4\sin \theta \]Show All Steps Hide All Steps
Start SolutionThere really isn’t a whole lot to do here. All we need to do is to use the following conversion formulas in the equation where (and if) possible
\[\begin{array}{c}x = \rho \sin \varphi \cos \theta \hspace{0.5in}y = \rho \sin \varphi \sin \theta \hspace{0.5in}z = \rho \cos \varphi \\ {\rho ^2} = {x^2} + {y^2} + {z^2}\end{array}\] Show Step 2To make this problem a little easier let’s first do some rewrite on the equation.
First, let’s deal with the cosecant.
\[\frac{1}{{\sin \varphi }} = 2\cos \theta + 4\sin \theta \hspace{0.25in}\,\,\, \to \hspace{0.5in}1 = 2\sin \varphi \cos \theta + 4\sin \varphi \sin \theta \]Next, let’s multiply everything by \(\rho \) to get,
\[\rho = 2\rho \sin \varphi \cos \theta + 4\rho \sin \varphi \sin \theta \]Doing this makes recognizing the terms on the right a little easier.
Show Step 3Using the appropriate conversion formulas from Step 1 gives,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {{x^2} + {y^2} + {z^2}} = 2x + 4y}}\]