In an RLC circuit, the phase relationship between the voltage and current is shown on an oscilloscope as the circuit passes through resonance. By varying the inductance or capacitance, the phase difference between the voltage across the inductor, capacitor, and resistor is observed. At resonance, the voltage and current are in phase, while off-resonance, the circuit exhibits either a leading or lagging phase shift depending on whether the circuit is more capacitive or inductive. This illustrates the dynamic phase relationships in RLC circuits and the concept of resonance.