Answer
\[K = \frac{{ab}}{{{{\left( {\sqrt {{a^2}{{\sin }^2}\omega t{\mathbf{i}} + {b^2}{{\cos }^2}\omega t{\mathbf{j}}} } \right)}^3}}}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = a\cos \omega t{\mathbf{i}} + b\sin \omega t{\mathbf{j}} \hfill \\
{\text{By Theorem 12}}{\text{.8 }} \hfill \\
{\text{If }}C{\text{ is a smooth curve given by }}{\mathbf{r}}\left( t \right),{\text{ then the curvature }}K{\text{ of}} \hfill \\
C{\text{ at }}t{\text{ is }}K = \frac{{\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\mathbf{r}}'\left( t \right)} \right\|}^3}}} \hfill \\
{\mathbf{r}}\left( t \right) = a\cos \omega t{\mathbf{i}} + b\sin \omega t{\mathbf{j}} \hfill \\
{\mathbf{r}}'\left( t \right) = - a\omega \sin \omega t{\mathbf{i}} + b\omega \cos \omega t{\mathbf{j}} \hfill \\
{\mathbf{r}}''\left( t \right) = - a{\omega ^2}\cos \omega t{\mathbf{i}} - b{\omega ^2}\sin \omega t{\mathbf{j}} \hfill \\
\hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{ - a\omega \sin \omega t}&{b\omega \cos \omega t}&0 \\
{ - a{\omega ^2}\cos \omega t}&{ - b{\omega ^2}\sin \omega t}&0
\end{array}} \right| \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{b\omega \cos \omega t}&0 \\
{ - b{\omega ^2}\sin \omega t}&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{ - a\omega \sin \omega t}&0 \\
{ - a{\omega ^2}\cos \omega t}&0
\end{array}} \right|{\mathbf{j}} \hfill \\
+ \left| {\begin{array}{*{20}{c}}
{ - a\omega \sin \omega t}&{b\omega \cos \omega t} \\
{ - a{\omega ^2}\cos \omega t}&{ - b{\omega ^2}\sin \omega t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left( {ab{\omega ^3}{{\sin }^2}\omega t + ab{\omega ^3}{{\cos }^2}\omega t} \right){\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = ab{\omega ^3}\left( {{{\sin }^2}\omega t + {{\cos }^2}\omega t} \right){\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = ab{\omega ^3}{\mathbf{k}} \hfill \\
\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\| = ab{\omega ^3} \hfill \\
and \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = \left\| { - a\omega \sin \omega t{\mathbf{i}} + b\omega \cos \omega t{\mathbf{j}}} \right\| \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = \sqrt {{a^2}{\omega ^2}{{\sin }^2}\omega t{\mathbf{i}} + {b^2}{\omega ^2}{{\cos }^2}\omega t{\mathbf{j}}} \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = \omega \sqrt {{a^2}{{\sin }^2}\omega t{\mathbf{i}} + {b^2}{{\cos }^2}\omega t{\mathbf{j}}} \hfill \\
{\text{Therefore,}} \hfill \\
K = \frac{{\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\mathbf{r}}'\left( t \right)} \right\|}^3}}} = \frac{{ab{\omega ^3}}}{{{\omega ^3}{{\left( {\sqrt {{a^2}{{\sin }^2}\omega t{\mathbf{i}} + {b^2}{{\cos }^2}\omega t{\mathbf{j}}} } \right)}^3}}} \hfill \\
K = \frac{{ab}}{{{{\left( {\sqrt {{a^2}{{\sin }^2}\omega t{\mathbf{i}} + {b^2}{{\cos }^2}\omega t{\mathbf{j}}} } \right)}^3}}} \hfill \\
\end{gathered} \]
