Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 2

In the cylindrical coordinate system, $r=\sqrt{x^2+y^2}=5$ and $z=-1$. The point is in the third quadrant, so a representative azimuth in $[0,2\pi )$ is $\pi +\arctan (4/3)$. ThereforeThe angle is defined modulo $2\pi$.

For spherical coordinates, the distance from the origin is $\sqrt{26}$ and the polar angle satisfies $\cos \theta =z/r=-1/\sqrt{26}$. The azimuth is the same as in part (a). ThusThe polar angle is in $(\pi /2,\pi )$ because the point lies below the $xy$-plane.