Question

Solve this problem in the png file question

Answer 1

According to the diagram,
`\theta` is the polar angle (the "vertical" angle made with the positive z-axis) and
`\phi` is the azimuthal angle (the "horizontal" angle made with the positive x-axis), so the convention used here is to take

`x^2 + y^2 + z^2 = r^2 \\\\ x = r \cos(\phi) \sin(\theta) \\\\ y = r \sin(\phi) \sin(\theta) \\\\ z = r \cos(\phi)`

Then for the spherical point (1, π/4, π/2), we have the corresponding Cartesian point (x, y, z), where

`x = 1 \cos\left(\frac\pi4\right) \sin\left(\frac\pi2\right) = \frac1{\sqrt2} \\\\ y = 1 \sin\left(\frac\pi4\right) \sin\left(\frac\pi2\right) = \frac1{\sqrt2} \\\\ z = 1 \cos\left(\frac\pi2\right) = 0`