A particle moves in such a way that its position vector varies with time as $\overrightarrow{\mathrm{r}}=\mathrm{Acos\omega t}\widehat{\mathrm{i}}+\mathrm{Asin\omega t}\widehat{\mathrm{j}}$. The initial velocity of the particle is

Substituting $\overrightarrow{\mathrm{r}}=\left(\mathrm{Acos\omega t}\widehat{\mathrm{i}}+\mathrm{Asin\omega t}\widehat{\mathrm{j}}\right)$

$\mathrm{In}\overrightarrow{\mathrm{v}}=\frac{\mathrm{d}\overrightarrow{\mathrm{r}}}{\mathrm{dt}}\mathrm{we}\mathrm{have}\overrightarrow{\mathrm{v}}=\mathrm{A}\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{cos\omega t}\right)\widehat{\mathrm{i}}+\mathrm{A}\frac{\mathrm{d}}{\mathrm{dt}}\left(\sin \mathrm{\omega t}\right)\widehat{\mathrm{j}}$

                                               $=-\mathrm{A\omega sin\omega t}\widehat{\mathrm{i}}+\mathrm{A\omega cos\omega t}\widehat{\mathrm{j}}$

$\mathrm{At}\mathrm{t}=0,\mathrm{v}=-\mathrm{A\omega sin}0\widehat{\mathrm{i}}+\mathrm{A\omega cos}0\widehat{\mathrm{j}}=\mathrm{A\omega }\widehat{\mathrm{j}}$