Question

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.$x=\sin ht,y=\cos ht,t\in R$

Short answer

The Curve plot

Step-by-step solution

Given information

The parametric curves,$x=\sin ht,y=\cos ht,t\in R$

Calculation

The goal is to draw the parametric curve.

Assume $t=-2,-1,0,1,2$when drawing the graph for the parametric equations.

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

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

$(x,y)=(\sinh t,\cosh t)$

$(x,y)=\left(\sinh \right(-2),\cosh (-2\left)\right)[$since substituting $t=-2]$

$(x,y)=(-3.62,3.76)$simplify

The point $(x,y)$When $t=-1$is,

$$\begin{gathered} (x,y)=(\sinh t,\cosh t) \\ (x,y)=\left(\sinh \right(-1),\cosh (-1\left)\right)[\text{since by substituting}t=-1] \\ (x,y)=(-1.17,1.54) \end{gathered}$$

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

$$\begin{gathered} (x,y)=(\sinh t,\cosh t) \\ (x,y)=\left(\sinh \right(0),\cosh (0\left)\right)\text{[since by substituting}t=0\left] \\ \right(x,y)=(0,1)\text{simplify} \end{gathered}$$

The point $(x,y)$When $t=1$ is,

$$\begin{gathered} (x,y)=(\sinh t,\cosh t) \\ (x,y)=\left(\sinh \right(1),\cosh (1\left)\right)[\text{since by substituting}t=1] \\ (x,y)=(1.17,1.54) \end{gathered}$$

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

$$\begin{gathered} (x,y)=(\sinh t,\cosh t) \\ (x,y)=\left(\sinh \right(2),\cosh (2\left)\right)[\text{since by substituting}t=2] \\ (x,y)=(3.62,3.76)\text{simplify} \end{gathered}$$

Calculation

The points are represented in tabular form as follows:

The following is a graphical representation of the situation:

Therefore, the solution is the required graph.