According to the diagram, [tex]\theta[/tex] is the polar angle (the "vertical" angle made with the positive z-axis) and [tex]\phi[/tex] is the azimuthal angle (the "horizontal" angle made with the positive x-axis), so the convention used here is to take
[tex]x^2 + y^2 + z^2 = r^2 \\\\ x = r \cos(\phi) \sin(\theta) \\\\ y = r \sin(\phi) \sin(\theta) \\\\ z = r \cos(\phi)[/tex]
Then for the spherical point (1, π/4, π/2), we have the corresponding Cartesian point (x, y, z), where
[tex]x = 1 \cos\left(\dfrac\pi4\right) \sin\left(\dfrac\pi2\right) = \dfrac1{\sqrt2} \\\\ y = 1 \sin\left(\dfrac\pi4\right) \sin\left(\dfrac\pi2\right) = \dfrac1{\sqrt2} \\\\ z = 1 \cos\left(\dfrac\pi2\right) = 0[/tex]
