Two simple harmonic motion, are represented by the equations ${y}_{1}=10 \sin \left(3 \pi {t}+\frac{\pi}{3}\right)$
$y_{2}=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t)$
Ratio of amplitude of ${y}_{1}$ to ${y}_{2}={x}: 1$. The value of ${x}$ is ...... .
JEE MAIN 2021, Medium
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${y}_{1}=10 \sin \left(3 \pi {t}+\frac{\pi}{3}\right) \Rightarrow \text { Amplitude }=10$
${y}_{2}=5(\sin 3 \pi {t}+\sqrt{3} \cos 3 \pi {t})$
${y}_{2}=10\left(\frac{1}{2} \sin 3 \pi {t}+\frac{\sqrt{3}}{2} \cos 3 \pi {t}\right)$
${y}_{2}=10\left(\cos \frac{\pi}{3} \sin 3 \pi {t}+\sin \frac{\pi}{3} \cos 3 \pi {t}\right)$
${y}_{2}=10 \sin \left(3 \pi {t}+\frac{\pi}{3}\right) \Rightarrow \text { Amplitude }=10$
So ratio of amplitudes $=\frac{10}{10}=1$
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