differentiation help

Question from IsaacPhysics:
The displacement x of a damped, simple harmonic oscillator as a function of time t is given by x=Ae^(−αt) cos(ωt+ϕ). Find a general expression for the times at which the displacement is a maximum or minimum. (Your answer will involve an inverse trig function and an arbitrary integer n.)

My (wrong) answer: t = (arctan( - (alpha)/(omega)) - phi)/(omega)
I differentiated the expression for x with respect to t, and set dx/dt = 0 and rearranged for t.
I'm not sure why it's wrong or where to get the 'arbitrary integer n' from?

Reply 1

mqb2766

Looks about right though it would be handy to have the question link to check. The way youve written it is that there is a single solution (corresponding to arctan) whereas obviously there are muliptle solutions corresponding to the "+ n*pi".