(Solved): Find an equation for the circle of curvature of the curve \( \mathbf{r}(\math ...

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Find an equation for the circle of curvature of the curve \( \mathbf{r}(\mathrm{t})=8 \mathrm{t} i+\sin (5 \mathrm{t}) \mathbf{j} \) at the point \( (4 \pi, 1) \). (The curve parameterizes the graph of \( y=\sin \left(\frac{5}{8} x\right) \) in the \( x y- \) plane. \( ) \) An equation for the circle of curvature is (Type an equation. Type an exact answer, using \( \pi \) as needed.)