The equation of displacement of two waves are given as ${y_1} = 10\sin \left( {3\pi t + \frac{\pi }{3}} \right)$; ${y_2} = 5(\sin 3\pi t + \sqrt 3 \cos 3\pi t)$. Then what is the ratio of their amplitudes
AIIMS 1997, Medium
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(c) ${y_1} = 10\sin \,\left( {3\pi t + \frac{\pi }{3}} \right)$ ...$(i)$
and ${y_2} = 5[\sin 3\pi t + \sqrt 3 \cos \,3\pi t]$
$ = 5 \times 2\,\left[ {\frac{1}{2} \times \sin 3\pi t + \frac{{\sqrt 3 }}{2} \times \cos 3\pi t} \right]$
$ = 10\,\left[ {\cos \frac{\pi }{3}\sin 3\pi t + \sin \frac{\pi }{3}\cos \pi t} \right]$
$ = 10\,\left[ {\sin \,\left( {3\pi t + \frac{\pi }{t}} \right)} \right]$ ...$ (ii)$ ( $\because$ $sin(A + B) = sinA \,cosB + cosA sinB)$
Comparing equation $ (i)$ and $(ii)$ we get ratio of amplitude $1 : 1.$
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