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The spherical coordinates of (-3, 4, -12) are (x, y, z). Find tan y + tan z.

 May 6, 2019
 #1
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We don't use (x,y,z) for spherical coordinates.  It just causes confusion.

 

\(\text{I'm going to assume that }(x,y,z) \equiv (\rho, \theta, \phi)\\ \begin{align*} -3 &= \rho \sin(\theta)\cos(\phi)\\ 4 &= \rho\sin(\theta)\sin(\phi)\\ -12 &= \rho\cos(\theta) \end{align*}\)

 

\(\theta= \arccos\left(\dfrac{-12}{\sqrt{(-3)^2+4^2+(-12)^2}}\right) = \arccos\left(\dfrac{-12}{13}\right)\\ \tan(\theta)=-\dfrac{\sqrt{13^2-12^2}}{12} = -\dfrac{5}{12}\)

 

\(\phi= \arctan\left(\dfrac{4}{-3}\right)\\ \tan(\phi)= -\dfrac 4 3\)

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 May 7, 2019

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