Converting $(0, -6, 0)$ from rectangular coordinates to spherical. In particular, finding $\theta$

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My homework asks me to convert the point $(0, -6, 0)$ from rectangular to spherical coordinates. I've found that $\rho = 6$ and $\phi = \frac{\pi}{2}$, but I'm stuck on determining $\theta$.

If $x = \rho sin(\phi)cos(\theta)$ and $y = \rho sin(\phi)sin(\theta)$, then $\theta = arctan(\frac{y}{x})$, but x is zero in this case, and division by zero is undefined. Have I made a mistake somewhere?

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1 Answer

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(Turning my comment into an answer.)

Conversion formulas aside, $\theta$ measures the angle in the $x$-$y$ plane from the positive $x$-axis. The given point is on the negative $y$-axis, so depending on the convention you’re using for $\theta$’s allowed values, $\theta$ is either $-\frac\pi2$ or $\frac{3\pi}2$.

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