Answer
$$\kappa \left( t \right) = \frac{1}{a}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left\langle {a\sin t,a\cos t} \right\rangle \cr
& {\text{The parametrc curve is }}{\bf{r}}\left( t \right) = \left\langle {f\left( t \right),g\left( t \right)} \right\rangle \cr
& f\left( t \right) = a\sin t \cr
& g\left( t \right) = a\cos t \cr
& {\text{Calculate the first and second derivatives of }}f\left( t \right){\text{ and }}g\left( t \right) \cr
& f'\left( t \right) = a\cos t \cr
& g'\left( t \right) = - a\sin t \cr
& f''\left( t \right) = - a\sin t \cr
& g''\left( t \right) = - a\cos t \cr
& {\text{Use the formula }}\kappa \left( t \right) = \frac{{\left| {f'g'' - f''g'} \right|}}{{{{\left( {{{\left( {f'} \right)}^2} + {{\left( {g'} \right)}^2}} \right)}^{3/2}}}} \cr
& \kappa \left( t \right) = \frac{{\left| {\left( {a\cos t} \right)\left( { - a\cos t} \right) - \left( { - a\sin t} \right)\left( { - a\sin t} \right)} \right|}}{{{{\left( {{{\left( {a\cos t} \right)}^2} + {{\left( { - a\sin t} \right)}^2}} \right)}^{3/2}}}} \cr
& \kappa \left( t \right) = \frac{{\left| { - {a^2}{{\cos }^2}t - {a^2}{{\sin }^2}t} \right|}}{{{{\left( {{a^2}{{\cos }^2}t + {a^2}{{\sin }^2}t} \right)}^{3/2}}}} \cr
& \kappa \left( t \right) = \frac{{\left| { - {a^2}} \right|}}{{{{\left( {{a^2}} \right)}^{3/2}}}} = \frac{{{a^2}}}{{{a^3}}} \cr
& \kappa \left( t \right) = \frac{1}{a} \cr} $$
