Amplitude of a harmonic oscillator

The displacement of a harmonic oscillator is given by \(x = \alpha \sin\omega t + \beta \cos\omega t\). The amplitude of the oscillation is

\(\alpha + \beta\)

\(\alpha^2 + \beta^2\)

\(\sqrt{\alpha^2 + \beta^2}\)

\(\sqrt{\alpha^2 + \beta^2 + 2\alpha\beta}\)

Since the two perpendicular components have a phase difference of \(\frac{\pi}{2}\), the net amplitude is \(A = \sqrt{\alpha^2 + \beta^2}\).