Q 3. - Calculus

Question

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

Step-by-Step Solution

Answer

The graph 

Step 1: Given information

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

Step 2: 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,

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

Step 3: Calculation

The tabular representation of the points is as follows, 

The graphical representation is shown below, 

Therefore, the solution is the required graph.