How to write basis vectors in Cartesian coordinates in terms of cylindrical coordinates?

Question

Given a Cartesian coordinate system with basis vectors (e_x, e_y, e_z)

and a Cylindrical coordinate system with basis vectors (e_r, e_θ, e_z)

Why and how does:

e_x = e_rCos(θ) - e_θSin(θ) ?

e_y = e_rSin(θ) + e_θCos(θ) ?

This is from a derivation of the del operator in cylindrical coordinates from my lecture notes.

I cannot for the life of me figure this out. Shouldn't e_x = e_rCos(θ) and e_y = e_rSin(θ)?

What is it with the extra Sin and Cos?

A simple, detailed derivation/ explanation would be very helpful.