Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
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,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {{x^2} + {y^2} + {z^2}} = 2x + 4y}}\]