>>11210880because the voltage over and current through a capacitor are related through the relation .
When you have include an inductor in the circuit and apply kirchoff's laws, you get a system than can be described by a second order linear ordinart differential equation. SOLODEs can be solved by using something called a characteristic equation, which turns out to be a quadratic equation. If the discriminant of this quadratic is less than 0, you get that the current is a function of an exponential with a complex argument. Complex arguments in exponential functions => sinusoidal behavior. That's why.
