Question

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve. $x=\sin t,y=\cos t,t\in [0,4\pi ]x=\sin t,y=\cos t,t\in [0,4\pi ]$

Short answer

The graph

Step-by-step solution

Given information

The parametric curves, $x=\sin t,y=\cos t,t\in [0,4\pi ]$

Calculation

The goal is to draw the parametric curve.

Assume that you want to draw a graph for the parametric equations $t=0,\frac{\pi }{2},\pi ,2\pi ,4\pi$

Find the values of $x,y$by substituting different $t$values in the parametric equations.

The point $(x,y)$When $t=0$is,

$(x,y)=(\sin t,\cos t)[$since by substituting $t=0]$

$(x,y)=(\sin 0,\cos 0)$simplify

$(x,y)=(0,1)$

The point $(x,y)$When $t=\frac{\pi }{2}$is,

$$\begin{gathered} (x,y)=(\sin t,\cos t) \\ \left.(x,y)=\left(\sin \frac{\pi }{2},\cos \frac{\pi }{2}\right)\text{[since by substituting}t=1\right] \\ (x,y)=(1,0)\text{simplify} \end{gathered}$$

The point $(x,y)$When $t=\pi$is,

$$\begin{gathered} (x,y)=(\sin t,\cos t) \\ (x,y)=(\sin \pi ,\cos \pi )[\text{since by substituting}t=\pi ] \\ (x,y)=(0,-1)\text{simplify} \end{gathered}$$

The point $(x,y)$When $t=2\pi$is,

$$\begin{gathered} (x,y)=(\sin t,\cos t) \\ (x,y)=(\sin 2\pi ,\cos 2\pi )[\text{since by substituting}t=2\pi ] \\ (x,y)=(0,1)\text{simplify} \end{gathered}$$

The point $(x,y)$ When $t=4\pi$ is,

$$\begin{gathered} (x,y)=(\sin t,\cos t) \\ (x,y)=(\sin 4\pi ,\cos 4\pi )[\text{since by substituting}t=1] \\ (x,y)=(0,1)\text{simplify} \end{gathered}$$

Calculation

The tabular representation of the points is as follows,

The graphical representation is shown below,

Therefore, the solution is the required graph.