Four simple harmonic vibrations:

${y_1} = 8\,\cos\, \omega t;\,{y_2} = 4\,\cos \,\left( {\omega t + \frac{\pi }{2}} \right)$ ; 

${y_3} = 2\cos \,\left( {\omega t + \pi } \right);\,{y_4} = \,\cos \,\left( {\omega t + \frac{{3\pi }}{2}} \right)$ , 

are superposed on each other. The resulting amplitude and phase are respectively;

  • A$\sqrt {45} $ and ${\tan ^{ - 1}}(1/2)$
  • B$\sqrt {45} $ and  ${\tan ^{ - 1}}(1/3)$
  • C$\sqrt {75} $ and ${\tan ^{ - 1}}(1/2)$
  • D$\sqrt {75} $ and  ${\tan ^{ - 1}}(1/3)$
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