Question

In Exercises \(47-52,\) convert the point from spherical coordinates to cylindrical coordinates. $$ (6,-\pi / 6, \pi / 3) $$

Short answer

The cylindrical coordinates for the given point in spherical coordinates are \((3\sqrt{3}, -\pi / 6, 3)\)

Step-by-step solution

Identify the given coordinates

The given spherical coordinates are \(r = 6\), \(\theta = -\pi / 6\), and \(\phi = \pi / 3\).

Convert from spherical to cylindrical

Use the following conversion formulas to convert from spherical to cylindrical coordinates: \[\rho = r \sin(\phi)\]\[z = r \cos(\phi)\]\[\phi = \theta\]

Substitute the given values

Substitute \(r = 6\), \(\theta = -\pi / 6\), and \(\phi = \pi / 3\) into the formulas for \(\rho\) and \(z\), and use \(\theta\) as \(\phi\): \[\rho = 6 \sin(\pi / 3) = 3\sqrt{3}\]\[z = 6 \cos(\pi / 3) = 3\]\[\phi = -\pi / 6\]

Write the answer in cylindrical coordinates

So, the cylindrical coordinates for the given point in spherical coordinates are \((3\sqrt{3}, -\pi / 6, 3)\)