Question

In the given figure, if $i_{1}=3 \sin \omega t$ and $i_{2}=4 \cos \omega t$, then $i_{3}$ is $5 \sin \left(\omega t+\phi^{\circ}\right)$. Find $\phi$.

Answer 1

$i_{3} =i_{1}+i_{2} $
$=5\left(\frac{3}{5} \sin \omega t+\frac{4}{5} \cos \omega t\right) $
$=5\left(\sin \omega t \cdot \cos 53^{\circ}+\cos \omega t \cdot \sin 53^{\circ}\right)$
$=5 \sin \left(\omega t+53^{\circ}\right) $