Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

Question

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.

Answer 1

The factor $Ae^{\alpha t}$ is never zero (unless $A=0$), so you are actually looking for the roots of $\cos (\omega t+\phi)$

But

$$\cos (\omega t+\phi)=0 \iff \omega t+\phi=\frac{\pi}2+k\pi, k\in\Bbb Z$$

Answer 2

If $\alpha, t, \omega, \phi$ are all real numbers and $A\neq 0$ (if $A=0$, then $i(t)=0$ for all $t$), then $$i(t)=0\iff \cos(\omega t +\phi) = 0,$$ since $e^{\alpha t}$ is never $0$. From here on, it should be simple to find the values of $t$.