Q. 1TB - Calculus

Question

Parametric equations for the unit circle: Find parametric equations for the unit circle centered at the origin of the xy-plane that satisfy the given conditions.

The graph is traced counterclockwise once on the interval [0, 2π] starting at the point (1, 0).

Step-by-Step Solution

Answer

The parametric equation for the unit circle centered at the origin of the xy-plane that satisfy the given condition is $\left.\begin{array}{r}x=\cos t\\ y=\sin t\end{array}\right\}0\le t\le 2\mathrm{\pi }.$

Step 1. Given Information.

It is given that the unit circle is centered at the origin and the graph is traced counterclockwise with a starting point $\left(1,0\right).$

Step 2. Find the parametric equation for the unit circle.

It is given that the circle is centered at the origin and it is a unit circle. Thus, $x^2+y^2=1 \mathrm{and} \left(x,y\right)=\left(0,0\right).$

Now, the parametric equation of a circle of center $\left(0,0\right)$ and radius $1$ is:

$$\begin{gathered} x\left(t\right)=\mathrm{co}s t \\ y\left(t\right)=\sin t \end{gathered}$$

Now, the graph is traced counterclockwise once with a starting point$\left(1,0\right),$

$$\begin{gathered} x\left(t\right)=\cos t \\ y\left(t\right)=\sin t \end{gathered}$$

Thus, the parametric equation of the circle is:

$\left.\begin{array}{r}x=\cos t\\ y=\sin t\end{array}\right\}0\le t\le 2\mathrm{\pi }.$