Cylindrical Coordinates — Question 1

Question 1

Convert the rectangular point $(-3,3\sqrt 3,5)$ to cylindrical coordinates using the convention $r\ge 0$ and $0\le\theta<2\pi$. Verify by converting back.

Question 1 – Solution

Strategy Compute $r$ first, then place the angle in the correct quadrant.

Conversion $$r=\sqrt{(-3)^2+(3\sqrt 3)^2}=6.$$ Because $x<0$ and $y>0$, the point lies in Quadrant II. With $\tan\theta=-\sqrt 3$, $\theta=2\pi/3$. Hence $$\boxed{(r,\theta,z)=\left(6,\frac{2\pi}{3},5\right)}.$$

Verification $x=6\cos(2\pi/3)=-3$, $y=6\sin(2\pi/3)=3\sqrt 3$, and $z=5$.