Convert to Polar Form...1

Discussion in 'Geometry and Trigonometry' started by nycmathguy, Mar 11, 2022.

  1. nycmathguy

    nycmathguy

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    Can you show me how to do 22 and 24?


    Screenshot_20220311-155742_Samsung Notes.jpg
     
    nycmathguy, Mar 11, 2022
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  2. nycmathguy

    MathLover1

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    22. x^2+4x+y^2+4y=0

    The polar coordinates r (the radial coordinate) and θ (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by
    x = r cos θ
    y = r sin θ, where r is the radial distance from the origin, and θ is the counterclockwise angle from the x-axis. In terms of x and y, r = sqrt(x^2 + y^2)
    θ = |tan^(-1)(y/x).

    so, substitute x = r cos θ ad y = r sin θ in given equation

    ( r cos (θ) )^2+4( r cos( θ) )+(r sin (θ))^2+4(r sin (θ))=0

    r^2 cos ^2(θ) +4r cos (θ) +r^2 sin ^2(θ)+4r sin( θ)=0

    r^2( cos ^2(θ) +sin ^2(θ))+4r cos θ +4r sin θ=0

    r^2*1+4r cos θ +4r sin θ=0

    r(r+4cos θ +4sin θ)=0


    24. x^2(x^2+y^2)=y^2

    (r cos (θ) )^2((r cos (θ) )^2+(r sin θ)^2)=(r sin θ)^2

    r^2cos ^2(θ) (r^2cos ^2(θ) +r^2 sin ^2(θ))=r^2 sin ^2
    r^2cos ^2(θ) (r^2(cos ^2(θ) +sin ^2(θ)))=r^2 sin ^2(θ)
    r^2cos ^2(θ) (r^2*1)=r^2 sin ^2(θ)
    r^2cos ^2(θ) r^2=r^2 sin ^2(θ)
    r^4cos ^2(θ) =r^2 sin ^2(θ).........simplify, divide by r
    r^3cos ^2(θ) = r sin ^2(θ)
     
    MathLover1, Mar 12, 2022
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  3. nycmathguy

    nycmathguy

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    Thank you.
     
    nycmathguy, Mar 12, 2022
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