A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, $x_1\left(t\right)=A \sin \omega t$ and $x_2\left(t\right)=A\sin \left(\omega t+\frac{4\pi }{3}\right).$ Adding a third sinusoidal displacement 

$x_3\left(t\right)=B\sin (\omega t+\varphi )$ brings the mass to a complete rest. The values of B and $\varphi$ are

Since the mass is brought to rest, the total displacement is zero, i.e. 

$\begin{array}{l}{x}_1\left(t\right)+x_2\left(t\right)+x_3\left(t\right)=0\\ \Rightarrow A\sin \omega t+A\sin \left(\omega t+\frac{4\pi }{3}\right)+B\sin (\omega t+\varphi )=0\end{array}$

Using $\sin \alpha +\sin \beta =2A\sin \left(\omega t+\frac{2\pi }{3}\right)\cos \frac{2\pi }{3}$

We get

$\begin{array}{l}2A\sin \left(\omega t+\frac{2\pi }{3}\right)\cos \frac{2\pi }{3}+B\sin (\omega t+\varphi )=0\\ \Rightarrow -A\sin \left(\omega t+\frac{2\pi }{3}\right)+B\sin (\omega t+\varphi )+0 \left(\because \cos \frac{2\pi }{3}=-\frac{1}{2}\right)\end{array}$

$\Rightarrow B\sin (\omega t+\varphi )=A\sin \left(\omega t+\frac{2\pi }{3}\right)$

which gives $B=A\text{ and }\varphi =\frac{2\pi }{3}$