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Section 12.13 : Spherical Coordinates

6. Convert the equation written in Spherical coordinates into an equation in Cartesian coordinates.

cscφ=2cosθ+4sinθ

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Start Solution

There 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

x=ρsinφcosθy=ρsinφsinθz=ρcosφρ2=x2+y2+z2 Show Step 2

To make this problem a little easier let’s first do some rewrite on the equation.

First, let’s deal with the cosecant.

1sinφ=2cosθ+4sinθ1=2sinφcosθ+4sinφsinθ

Next, let’s multiply everything by ρ to get,

ρ=2ρsinφcosθ+4ρsinφsinθ

Doing this makes recognizing the terms on the right a little easier.

Show Step 3

Using 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}}