RMS amplitude

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Failed to parse (unknown function "\pift"): {\displaystyle y(t) = Asin (2\pift+\varphi) = Asin(\omegat+\varphi)}
Failed to parse (unknown function "\omegat"): {\displaystyle Y_RMS = \sqrt{\tfrac{1}{T} \int_{0}^{T} [Asin(\omegat)]^2 dt}}
Failed to parse (unknown function "\omegat"): {\displaystyle       = A\sqrt{\tfrac{1}{T} \int_{0}^{T} \tfrac{1-cos(2\omegat)}{2} dt}}
Failed to parse (unknown function "\omegat"): {\displaystyle       = A\sqrt{\tfrac{1}{T} [\tfrac{T}{2} - \tfrac{sin(2\omegat)}{4\omega}]_0^T}}
Failed to parse (syntax error): {\displaystyle x_RMS=\sqrt{\tfrac{1}{n}\sum_{i=1}^nx_i^2}                                                                     (1) The RMS of the corresponding formula for a continuous waveform f(t) defined over the interval [T1, T2] is f_{RMS}=\sqrt{\tfrac{T_2}{T_1}\textstyle \int_{T_1}^{T_2} \displaystyle [f(t)]^2}dt             (2) and the RMS for a function over all time is f_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [f(t)]^2}dt                                                                 (3) For a sine wave y(t)=A sin⁡(2πft+φ)=Asin(ωt+φ)                                                                                           (4) where y is displacement, t is time, f is frequency, and A is amplitude (the peak deviation of the function from zero) Y_{RMS}=\sqrt{\tfrac{1}{T}\int_{0}^{T} [Asin(\omegat)]^2dt}                =A\sqrt{\tfrac{1}{T}\int\limits_{0}^{T} \tfrac{1-cos(2\omegat)}{2}dt}                =A\sqrt{\tfrac{1}{T}[\tfrac{T}{2}-\tfrac{sin(2\omegat)}{4\omega}]_0^T}                =\tfrac{A}{\sqrt{2}}                                                                                                (5)  }