Question

Sketch the parametric curve by plotting points.

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

Short answer

The required graph is

Step-by-step solution

Given information

The parametric curve by plotting points.

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

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">x,y.

The point (x, y) When width="32">t=0is,

width="153">(x,y)=2sin3t,2cos3t

width="161">(x,y)=2sin30,2cos30[since by substituting width="42">t=0

width="145">x4y=(2·0,2·1)(x,y)=(0,2)simplify

The point width="38">(x,y)When width="44">t=π2is,

width="324">(x,y)=2sin3t,2cos3t(x,y)=2sin3π2,2cos3π2(x,y)=(2·1,2·0)since by substitutingt=π2(x,y)=(2,0)simplify

The point width="38">(x,y)When width="36">t=πis,

localid="1653374594977" $$\begin{gathered} (x,y)=\left(2{\sin }^{3}t,2{\cos }^{3}t\right) \\ (x,y)=\left(2{\sin }^3\pi ,2{\cos }^3\pi \right) \\ (x,y)=(2·0,2·-1) \\ (x,y)=(0,-2) \end{gathered}$$

Simplification

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

The tabular representation of the points is as follows,

The graphical representation is