If ${\mathrm{i}}_1=3\sin \mathrm{\omega t}\text{ and }{\mathrm{i}}_2=4\cos \mathrm{\omega t}\text{ then }{\mathrm{i}}_3=5\sin (\mathrm{\omega t}+\mathrm{\varphi })$. What is the value of $\text{'}\mathrm{\varphi }\text{'}.$ (in degrees)?

${\mathrm{i}}_3={\mathrm{i}}_1+{\mathrm{i}}_2=3\mathrm{sin\omega t}+4\mathrm{cos\omega t} \cdots \left(i\right)$.

Let, $3=5\mathrm{cos\varphi } \cdots \left(ii\right)$ and $4=5\mathrm{sin\varphi } \cdots \left(iii\right)$.

Using (ii) and (iii) in (i), ${\mathrm{i}}_3=5\sin \left(\mathrm{\omega t}+\mathrm{\varphi }\right)$.

Also from (ii) and (iii), 

$\mathrm{tan\varphi }=\frac{4}{3} \Rightarrow \mathrm{\varphi }={\tan }^{-1}\left(\frac{4}{3}\right)\approx 53°$.