x =\sin \left(\frac{\theta}{2}\right), y = \cos \left(\frac{\theta}{2}\right), -\pi \leq \theta \leq \pi (a) Eliminate the parameter to find a Cartesian equation of the curve. and how does thecurve look

lwfrgin

lwfrgin

Answered question

2021-05-08

x=sin(θ2),y=cos(θ2),πθπ (a) Eliminate the parameter to find a Cartesian equation of the curve. and how does thecurve look

Answer & Explanation

Liyana Mansell

Liyana Mansell

Skilled2021-05-09Added 97 answers

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-30Added 2605 answers

Consider the following parametric equations,

x=sin(θ2) and y=cos(θ2), πθπ

To eliminate θ, square x and y both sides and add them

x2=sin2(θ2)

y2=cos2(θ2)

x2+y2=sin2(θ2)+cos2(θ2)

x2+y2=1

The equation x2+y2=1 represents a circle of radius. Given that the perimeter θ varies from π to π

Substitute θ=π in the perimetric equations

x=sin(θ2) and y=cos(θ2)

x=sin(π2)=1

y=cos(π2)=0

So the point corresponding to θ=π is (1,0)

Substitute θ=π in the perimetric equations

x=sin(θ2) and y=cos(θ2)

x=sin(π2)=1

y=cos(π2)=0

So the point corresponding to θ=π is (1,0)

Substitute θ=π in the perimetric equations

x=sin(θ2) and y=cos(θ2)

x=sin(π2)=1

y=cos(π2)=0

So, the point corresponding to θ=π is (1,0)

Hence, the curve is a senincircle with origin (0,0) radius 1.

As θ increases from π to π, the point traces semicircles in clockwise direction, storting from the point (1,0) to the point (1,0)

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