Exercise 23

You might recall the pythagorean identity from trigonometry given by $$\begin{gathered} \cos^2(\theta)+\sin^2(\theta)=1 \end{gathered}$$ We want to use this identity to parameterize the circle with center $(h,k)$ and radius $r \gt 0$ given by $$\begin{gathered} (x-h)^2 + (y-k)^2 = r^2 \end{gathered}$$

1. Reorganize the general equation of a circle $$\begin{gathered} (x-h)^2 + (y-k)^2 = r^2 \end{gathered}$$ to be of the form $$\begin{gathered} (\underline{\hspace{2cm}})^2+(\underline{\hspace{2cm}})^2=1 \end{gathered}$$

2. Because identities work for any value, we can use the pythagorean identity to parameterize the general equation of a circle. Fill in the blanks below with each of the respective quantities from the previous task $$\begin{aligned} \cos(t) = & \underline{\hspace{2cm}} \\ \sin(t) = & \underline{\hspace{2cm}} \end{aligned}$$

3. Solve each of the two equations from the previous part for $x$ and $y$ to get parametric equations for a generalized circle.

4. Write a couple sentences about what $r$, $h$, and $k$ mean in the parametric equations for a generalized circle.