Question

Change from rectangular to cylindrical coordinates. Select such set of cylindrical coordinates as θ [0, 2π).

(3(3^(1/2)), 3, -5)

I've got r = 6 and z = -5. How do I find theta?
I keep getting it to be some variation of 30 degrees. :/

Answer 1

First, good job finding r = sqrt(27 + 9) and z = -5.

To find theta simply, try drawing a picture of the x-y (or r-theta) plane.
This should look like a familiar triangle: a right triangle made by half an equilateral, which confirms your "variation of 30 degrees" (though you'll want to put that into radians).

To confirm this conversion, use the following equations:
\[x = rsin\theta, y = rcos\theta \implies tan\theta = \frac{rsin\theta}{rcos\theta} \implies \theta = arctan\left(\frac{x}{y}\right) \] Keep in mind that tan(theta) = tan(theta + pi), so check theta against x and y for reasonableness.