Integrate the current to find charge as a function of time, then divide the charge by e (the magnitude of the electron charge) to find the number of electrons as a function of time.
Assuming i(t) is in units of amps, the number of electrons is
N(t) = (1/e) q(t) = (1/e) ∫i(t)dt = (1/e) ∫10costdt = (1/e)(10sint + c)
q(t0) = 0 ⇒ 10sint0 + c = 0 ⇒ c = -10sint0
N(t) = (1/e) (10sint - 10sint0)