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. :/

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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. :/

MIT 18.02 Multivariable Calculus, Fall 2007
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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.

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