Answer
The curvature at $s = \frac{\pi }{2}$ is $\kappa \left( {\frac{\pi }{2}} \right) = \frac{9}{{64}}$
Work Step by Step
We have $\left( {x\left( s \right),y\left( s \right)} \right) = \left( {\sin 3s,2\sin 4s} \right)$.
The derivatives are
$\left( {x'\left( s \right),y'\left( s \right)} \right) = \left( {3\cos 3s,8\cos 4s} \right)$
$\left( {x{\rm{''}}\left( s \right),y{\rm{''}}\left( s \right)} \right) = \left( { - 9\sin 3s, - 32\sin 4s} \right)$
Using Eq. (11) we compute the curvature
$\kappa \left( s \right) = \frac{{\left| {x'\left( s \right)y{\rm{''}}\left( s \right) - x{\rm{''}}\left( s \right)y'\left( s \right)} \right|}}{{{{\left( {x'{{\left( s \right)}^2} + y'{{\left( s \right)}^2}} \right)}^{3/2}}}}$
$\kappa \left( s \right) = \frac{{\left| { - 96\left( {\cos 3s} \right)\left( {\sin 4s} \right) + 72\left( {\cos 4s} \right)\left( {\sin 3s} \right)} \right|}}{{{{\left( {9{{\cos }^2}3s + 64{{\cos }^2}4s} \right)}^{3/2}}}}$
The curvature at $s = \frac{\pi }{2}$ is
$\kappa \left( {\frac{\pi }{2}} \right) = \frac{{\left| { - 72} \right|}}{{{{\left( {64} \right)}^{3/2}}}} = \frac{{72}}{{512}} = \frac{9}{{64}}$
