Discussion

If the instantaneous displacement of a wave is \(y = 2(\sin 2\pi t + \sqrt{3}\cos 2\pi t)\) cm, then the amplitude of the wave in cm, is

Answer: A

The amplitude is 4 cm.

A combination \(a\sin\theta + b\cos\theta\) is a single sinusoid of amplitude \(\sqrt{a^{2} + b^{2}}\). Here \(a = 2\) and \(b = 2\sqrt{3}\):

\[ \begin{aligned} A &= \sqrt{2^{2} + (2\sqrt{3})^{2}} = \sqrt{4 + 12} = 4 \\ y &= 4\sin(2\pi t + \tfrac{\pi}{3}) \end{aligned} \]

2 is just the factor outside the bracket, and adding the coefficients directly does not give an amplitude.

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