Q. 22 - Calculus

Question

Sketch the parametric curve by plotting points. 

$x=2{\sin }^{3}t,y=2{\cos }^{3}t,t\in [0,2\pi ]$

Step-by-Step Solution

Answer

The required graph is 

Step 1: Given information

The parametric curve by plotting points. 

$x=2{\sin }^{3}t,y=2{\cos }^{3}t,t\in [0,2\pi ]$

Step 2: Calculation

Consider the parametric curves $x=2{\sin }^{3}t,y=2{\cos }^{3}t,t\in [0,2\pi ]$.

The objective is to sketch the parametric curve.

To draw the graph for the parametric equations, assume $t=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$.

Substitute different $t$values in the parametric equations and find the values of width="33" style="max-width: none; vertical-align: -4px;" $x, y.$

The point (x, y) When width="32" style="max-width: none; vertical-align: -4px;" $t=0$ is,

width="153" style="max-width: none; vertical-align: -5px;" $(x,y)=\left(2{\sin }^{3}t,2{\cos }^{3}t\right)$

width="161" style="max-width: none; vertical-align: -5px;" $(x,y)=\left(2{\sin }^{3}0,2{\cos }^{3}0\right)$ [since by substituting width="42" style="max-width: none; vertical-align: -4px;" $\left.t=0\right]$

width="145" style="max-width: none; vertical-align: -26px;" $$\begin{gathered} \left(x_{4}y\right)=(2·0,2·1) \\ (x,y)=(0,2)\text{ simplify } \end{gathered}$$

The point width="38" style="max-width: none; vertical-align: -5px;" $(x, y)$ When width="44" style="max-width: none; vertical-align: -15px;" $t=\frac{\pi }{2}$is,

width="324" style="max-width: none; vertical-align: -70px;" $$\begin{gathered} (x,y)=\left(2{\sin }^{3}t,2{\cos }^{3}t\right) \\ (x,y)=\left(2{\sin }^3\frac{\pi }{2},2{\cos }^3\frac{\pi }{2}\right) \\ (x,y)=(2·1,2·0)\left[\text{ since by substituting }t=\frac{\pi }{2}\right] \\ (x,y)=(2,0)\text{ simplify } \end{gathered}$$

The point width="38" style="max-width: none; vertical-align: -5px;" $(x, y)$When width="36" style="max-width: none; vertical-align: -4px;" $t=\pi$ is,

$$\begin{gathered} (x,y)=\left(2{\sin }^{3}t,2{\cos }^{3}t\right) \\ (x,y)=\left(2{\sin }^3\pi ,2{\cos }^3\pi \right) \\ (x,y)=(2·0,2·-1) \\ (x,y)=(0,-2) \end{gathered}$$

Step 3: Simplification

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

The tabular representation of the points is as follows,

The graphical representation is