Cylindrical Coordinates — Question 2

Question 2 – Solution

Strategy. Use $x=r\cos\theta$, $y=r\sin\theta$; radial distance is $r$ when $r\ge 0$.

Step 1: Convert Since $\cos(7\pi/6)=-\sqrt 3/2$ and $\sin(7\pi/6)=-1/2$, $$\boxed{(x,y,z)=(-2\sqrt 3,-2,-2)}.$$

Step 2: Distances Distance to the $z$-axis is $\sqrt{x^2+y^2}=4$. Distance to the origin is $$\sqrt{x^2+y^2+z^2}=\sqrt{16+4}=\boxed{2\sqrt 5}.$$

Verification. $x^2+y^2=12+4=16=r^2$, confirming the conversion and axis distance.