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If you plug a voltmeter into a DC battery, you get a single rock-solid number like 9Β V9\text{ V}. But look at an AC mains signal on an oscilloscope, and the voltage never sits still β€” it constantly swings positive, passes through zero, and plunges negative.
To describe a moving waveform accurately, electronics engineers track three specific values: peak voltage, peak-to-peak voltage, and the exact instantaneous voltage at any split second.

Peak vs. Peak-to-Peak

The peak voltage (VpV_p or VmaxV_{\text{max}}) is the maximum voltage measured from the center zero-volt baseline to the crest of the wave. The peak-to-peak voltage (Vpβˆ’pV_{p-p}) is the total vertical span from the lowest trough all the way to the highest crest.
Because a standard sine wave is symmetrical around zero, the peak-to-peak voltage is always exactly double the peak voltage: Vpβˆ’p=2VpV_{p-p} = 2V_p
The same rule applies to current: peak current (IpI_p) is the maximum current from baseline, and peak-to-peak current is Ipβˆ’p=2IpI_{p-p} = 2I_p.
Peak values tell you the limits of the wave β€” but what if you need to know the exact voltage at precisely t=2.5Β mst = 2.5\text{ ms}?

Instantaneous Values and the Sine Equation

An instantaneous value (v(t)v(t) or i(t)i(t)) is the voltage or current at one specific instant in time. In the 1830s, Michael Faraday discovered that spinning a wire loop inside a magnetic field generates a voltage proportional to the sine of the rotation angle.
Because one complete cycle rotates through 2Ο€2\pi radians (360∘360^\circ) in one period (TT), we define angular frequency (Ο‰\omega) as the rate of rotation in radians per second: Ο‰=2Ο€f=2Ο€T\omega = 2\pi f = \frac{2\pi}{T}
Multiplying peak voltage by the sine of the angle at time tt gives the universal formula for instantaneous AC voltage: v(t)=Vpsin⁑(Ο‰t)=Vpsin⁑(2Ο€ft)v(t) = V_p \sin(\omega t) = V_p \sin(2\pi f t)