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Chapter 9: Q. 35 (page 721)

Finish Example 1 (b) by showing that the graph of the parametric equations x=cost,y=sint,t鈭�[蟺,2蟺] is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

Short Answer

Expert verified

The required graph is

Step by step solution

01

Given Information

The parametric equations , x=cost,y=sint,t鈭�[蟺,2pi].

02

Calculation

Consider the parametric equations ,x=cost,y=sint,t鈭�[蟺,2蟺].

The objective is to sketch the parametric curve by eliminating the parameter.

Take the equation: x=cost.

Squaring the equation on both sides.

x2=cos2t

Now take the equation y=sint.

y2=sin2t

Add the equations x2=cos2tand y2=sin2t.

Then, x2+y2=cos2t+sin2t

x2+y2=1sincecos2t+sin2t=1

The equation after eliminating the parameter t is x2+y2=1.

Now when t=蟺,

When t=2蟺

(x,y)=(cost,sint)(x,y)=(cos蟺,sin蟺)(x,y)=(-1,0)

Now when t=3蟺2,

(x,y)=(cost,sint)(x,y)=cos3蟺2,sin3蟺2(x,y)=(0,-1)
03

Further calculation

When t=2蟺

(x,y)=(cost,sint)(x,y)=(cos2蟺,sin2蟺)(x,y)=(1,0)

The graphical representation by using the points (-1,0)(0,-1)(1,0)is as follows,


The equation after elimination of the parameter is x2+y2=1.

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