RMS Value of AC

Because DC systems have a genuinely constant current or voltage, the term RMS -- meaning root mean square -- is specific to time-varying, sinusoidal AC quantities, and is not used at all in DC circuit analysis. The RMS value of an alternating current is defined as the square root of the mean of the squares of all the current values over one FULL cycle, denoted $I_{RMS}$ (with $V_{RMS}$ the corresponding definition for voltage).

Deriving this value for $i=I_m\sin\theta$: the squared current wave $i^2=I_m^2\sin^2\theta$ is entirely non-negative, so its sum over one full cycle can genuinely be averaged directly. Using the identity $\sin^2\theta=\tfrac12(1-\cos2\theta)$, the area under $i^2$ from $0$ to $2\pi$ is $\int_0^{2\pi}I_m^2\sin^2\theta\,d\theta = \dfrac{I_m^2}{2}\int_0^{2\pi}(1-\cos2\theta)\,d\theta = \dfrac{I_m^2}{2}(2\pi-0)=\pi I_m^2$ (the $\cos2\theta$ integral vanishes over a full period). Dividing by the full-cycle base length $2\pi$ gives the mean-square value $I_m^2/2$, and taking the square root gives

$I_{RMS} = \dfrac{I_m}{\sqrt2} \approx 0.707\,I_m \qquad (4.35)$ …