Question

Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.

$x=k\cos t,y=k\sin t$

Short answer

The curvature is$\frac{1}{k}.$

Step-by-step solution

Step 1. Given information.

The given function is$x=k\cos t,y=k\sin t$.

Step 2. Required answer.

Curvature is given by,

$$\begin{gathered} k=\frac{||T^{\text{'}}\left(t\right)||}{||r^{\text{'}}\left(t\right)||} \\ {r}^{\text{'}}\left(t\right)=⟨-k\sin t,k\cos t⟩ \\ ||r^{\text{'}}\left(t\right)||=\sqrt{(-k\sin t{)}^2+(k\cos t{)}^2} \\ =k \\ T\left(t\right)=\frac{r^{\text{'}}\left(t\right)}{||r^{\text{'}}\left(t\right)||} \\ =\frac{⟨-k\sin t,k\cos t⟩}{k} \\ =⟨-\sin t,\cos t⟩ \\ {T}^{\text{'}}\left(t\right)=⟨-\cos t,-\sin t⟩ \\ ||T^{\text{'}}\left(t\right)||=\sqrt{(-\cos t{)}^2+(-\sin t{)}^2}=1 \\ k=\frac{||T^{\text{'}}\left(t\right)||}{||r^{\text{'}}\left(t\right)||} \\ =\frac{1}{k} \end{gathered}$$