91Ó°ÊÓ

The parametric curves, $x=\sin t,y=\cos t,t∈[0,4\mathrm{π}]$

The goal is to draw the parametric curve.

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

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 $t=\frac{\mathrm{Ï€}}{2}$is,

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

The point $(x,y)$When $t=\mathrm{Ï€}$is,

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

The point $(x,y)$When $t=2\mathrm{Ï€}$is,

The point $(x,y)$ When $t=4\mathrm{Ï€}$ is,

The tabular representation of the points is as follows,

The graphical representation is shown below,

Therefore, the solution is the required graph.