Three waves represented by ${\mathrm{y}}_1=10\sin \left(50\mathrm{\pi t}-0.252\right)\mathrm{mm}{\mathrm{y}}_2=4\cos \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)\mathrm{mm}$ and ${\mathrm{y}}_3=7\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}+\mathrm{\pi }\right)\mathrm{mm}$ arrive at a given point. Then the resultant wave is represented by 

$\begin{array}{l}\mathrm{y}={\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3\\ =10\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)+4\cos \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)+7\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}+\mathrm{\pi }\right)\end{array}$

$=10\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)-7\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)+4\cos \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)$

$=3\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)+4\cos \left(50\mathrm{\pi t}-0.25\mathrm{x}\right)$

$\begin{array}{l}\Rightarrow \mathrm{y}=\sqrt{3^2+4^2}\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}+{\tan }^{-1}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\\ \Rightarrow \mathrm{y}=5\sin \left(50\mathrm{\pi t}-0.25\mathrm{x}+{\tan }^{-1}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\mathrm{mm}\end{array}$