Intercept between an Exponential and Sinusoidal Function I need to find the intercept point between a function $V_C=325.6 e^{t/0.22}$ and $V=325.6 \sin(2\pi\cdot 50t)$. I tried solving for $t$ in both equations and then solving when $V=V_C$ but i couldn't find a way to solve it. If it helps, the first equation, $V_C$, is a capacitor discharging, and the second one is a $230$V rms AC voltage source. They both start at the peak, $325.6$V, and I need to find out at what time will the voltage of the capacitor will be the same as the voltage source again. Thanks a lot.

The equation
$$
325.6\ e^{t/0.22}= 325.6 \sin(2\pi \cdot 50t)
$$
has no 'nice' exact solution. However, there are numeric techniques to finding the values $t$ that satisfy the equation. There are plenty of free calculators that will find intersections for you. For instance, try this link here. A few solutions are:
$$
\begin{aligned}
t &\approx -0.0138811 \\
t &\approx -0.016207 \\
t &\approx -0.0332926 \\
t &\approx -0.0367900
\end{aligned}
$$
