The problem:
Find the electric field a distance z from the center of a spherical surface of radius R (see figure) that carries a uniform charge density σ. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and θ. Be sure to take the positive square root: √ R2 + z 2 − 2Rz = (R − z) if R > z, but it is (z − R) if R < z.]
I understood the concept, but I just can't understand one little part in the solution below. They use that $cos \psi = \frac{z-Rcos\theta}{r}$ - why though? Why is it necessary/helpful?
$dq=\sigma R^2\sin(\theta)d\theta d\phi$
This is the total charge E. $E_z$ is the component along z-axis, which is at an angle of $\psi$ to the direction of E. Thus, multiplication with $\cos\psi$ is required.