# $S_1$ and $S_2$ are two sources of light which produce individually disturbance at a point P given by $\in _1=3 \sin wt,\;\in _2 =4 \cos wt$ Assuming $\overrightarrow {\in_1}$ and $\overrightarrow{\in_2}$ to be along the same line then result of their superposition is

$(a)\;5 \sin (wt+\tan^{-1} 4/3) \\ (b)\;5 \sin (wt) \\ (c)\;5 \sin [2wt+2 \tan^{-1} 4/3] \\ (d)\;5 \sin [ \tan^{-1} 4/3]$

$\in =\in _1+ \in _2 z$
$\quad= 3 \sin wt+ 4 \cos wt$
$\quad= 5 \bigg(\large\frac{3}{5}$$\sin wt +\large\frac{4}{5}$$ \cos wt \bigg)$
$\quad= 5 \sin (wt+\tan^{-1} 4/3)$
Hence a is the correct answer.

edited Jul 22, 2014