New Coordinate Systems

Polar coordinates describe the plane by distance and direction

Cylindrical coordinates stack height on polar coordinates

Spherical coordinates measure from the origin in one distance and two angles

Match the coordinate system to the symmetry of the problem

Convert the Cartesian point $(-3, 4)$ to polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$. Which answer is correct?

A point has cylindrical coordinates $\left(3, \tfrac{\pi}{2}, -2\right)$. Which Cartesian point does it name?

A point has spherical coordinates $\rho = 4$, $\phi = \tfrac{\pi}{3}$, $\theta = \tfrac{\pi}{6}$. Which Cartesian point does it name?

A solid region is described by $x^2 + y^2 \le 9$ and $0 \le z \le 4$. Which coordinate system makes the bounds of this region simplest?

Convert the Cartesian point $(2, 2, 5)$ to cylindrical coordinates and give just the radial coordinate $r$, in the form a*sqrt(b) with integers, like 2*sqrt(3).

Convert the Cartesian point $(0, 0, 5)$ to spherical coordinates and give just the polar angle $\phi$ measured from the positive $z$-axis, as a number of radians.