Convert the cylindrical coordinates (r, theta, z) = (2, 30 degrees, 5) to Cartesian coordinates.

Step-by-step solution

1. 1

   Recall the conversion equations. Cylindrical coordinates are polar coordinates in the $xy$-plane with an unchanged height:

   $x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=z$

   The third coordinate needs no work at all — the whole conversion happens in the plane.

2. 2

   Substitute the exact trigonometric values for $30^\circ$.

   $\cos 30^\circ=\frac{\sqrt3}{2},\qquad \sin 30^\circ=\frac12$

   Using exact surds rather than decimals keeps the answer exact.

3. 3

   Compute $x$ and $y$.

   $x=2\cdot\frac{\sqrt3}{2}=\sqrt3\approx 1.7321,\qquad y=2\cdot\frac12=1$

4. 4

   Carry the height across and state the point.

   $(x,y,z)=\left(\sqrt3,\,1,\,5\right)\approx(1.7321,\,1,\,5)$

5. 5

   Check by converting back. The radial distance is $\sqrt{x^2+y^2}=\sqrt{3+1}=2=r\;\checkmark$, and the angle is $\arctan\!\left(\tfrac{1}{\sqrt3}\right)=30^\circ\;\checkmark$ — in the first quadrant, as the positive $x$ and $y$ require. Both round-trip checks confirm the conversion.