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Sketch the parametric curve by plotting points.

x=2sin3t,y=2cos3t,tโˆˆ[0,2ฯ€]

Short Answer

Expert verified

The required graph is

Step by step solution

01

Given information

The parametric curve by plotting points.

x=2sin3t,y=2cos3t,tโˆˆ[0,2ฯ€]

02

Calculation

Consider the parametric curves x=2sin3t,y=2cos3t,tโˆˆ[0,2ฯ€].

The objective is to sketch the parametric curve.

To draw the graph for the parametric equations, assume t=0,ฯ€2,ฯ€,3ฯ€2,2ฯ€.

Substitute different tvalues 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=(2ยท0,2ยท1)(x,y)=(0,2)simplify

The point width="38">(x,y)When width="44">t=ฯ€2is,

width="324">(x,y)=2sin3t,2cos3t(x,y)=2sin3ฯ€2,2cos3ฯ€2(x,y)=(2ยท1,2ยท0)since by substitutingt=ฯ€2(x,y)=(2,0)simplify

The point width="38">(x,y)When width="36">t=ฯ€is,

localid="1653374594977" (x,y)=2sin3t,2cos3t(x,y)=2sin3ฯ€,2cos3ฯ€(x,y)=(2ยท0,2ยท-1)(x,y)=(0,-2)

03

Simplification

The point (x,y)When t=2ฯ€is,

The tabular representation of the points is as follows,

The graphical representation is

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