Q 7. Curve plot - Calculus

Question

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

Step-by-Step Solution

Answer

The Curve plot 

Step 1: Given information

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

Step 2: 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,

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

Step 3: 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.