The spherical coordinates of (-3, 4, -12) are (x, y, z). Find tan y + tan z.
+6248 We don't use (x,y,z) for spherical coordinates. It just causes confusion.
\(\text{I'm going to assume that }(x,y,z) \equiv (\rho, \theta, \phi)\\ \begin{align*} -3 &= \rho \sin(\theta)\cos(\phi)\\ 4 &= \rho\sin(\theta)\sin(\phi)\\ -12 &= \rho\cos(\theta) \end{align*}\)
\(\theta= \arccos\left(\dfrac{-12}{\sqrt{(-3)^2+4^2+(-12)^2}}\right) = \arccos\left(\dfrac{-12}{13}\right)\\ \tan(\theta)=-\dfrac{\sqrt{13^2-12^2}}{12} = -\dfrac{5}{12}\)
\(\phi= \arctan\left(\dfrac{4}{-3}\right)\\ \tan(\phi)= -\dfrac 4 3\)
