Finding cartesian equation from parametric trigonometric equationsI'm trying to find the cartesian equation of the curve which is defined parametrically by: x = 2 sin ⁡ θ , y = cos 2 ⁡ θBoth approaches I take result in the same answer: y = 1 − sin 2 ⁡ θ sin ⁡ θ = y − 1 x = 2 y − 1 x 2 = 4 ( y − 1 ) x 2 + 4 = 4 yMethod 2: sin 2 = y − 1 sin ⁡ θ = x 2 sin 2 ⁡ θ = x 2 4 x 2 + 4 = 4 yBut the answer listed is x 2 + 4 y 2 = 4. Are my calculations wrong?

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Finding cartesian equation from parametric trigonometric equationsI&#039;m trying to find the cartesian equation of the curve which is defined parametrically by:  x  =  2  sin  ⁡  θ  ,  y  =      cos    2    ⁡  θBoth approaches I take result in the same answer:  y  =  1  −      sin    2    ⁡  θ    sin  ⁡  θ  =      y    −    1      x  =  2      y    −    1          x    2    =  4  (  y  −  1  )        x    2    +  4  =  4  yMethod 2:      sin    2    =  y  −  1    sin  ⁡  θ  =      x    2          sin    2    ⁡  θ  =            x      2        4          x    2    +  4  =  4  yBut the answer listed is       x    2    +  4      y    2    =  4. Are my calculations wrong?

Claire Love · Accepted Answer

You have a mistake in your calculation. Note that      sin    2    ⁡  θ  ≠  y  −  1and that      sin    2    ⁡  θ  =  1  −  y  .By the way, I think you have a typo in your question.If   x  =  2  sin  ⁡  θ  ,  y  =      cos    2    ⁡  θ are correct, then since      sin    2    ⁡  θ  =            (              x        2            )        2    ,            cos    2    ⁡  θ  =  y  ,we have      cos    2    ⁡  θ  +      sin    2    ⁡  θ  =  1  ⇒  y  +            (              x        2            )        2    =  1  ⇒  y  =  −            x      2        4    +  1.Note that this is not       x    2    +  4      y    2    =  4

faois3nh · Accepted Answer

Answer: Squring each equation and multiplying the last on by 4 we get                              x          2                                    =        4        s        i                  n          2                θ                            4                  y          2                                    =        4        c        o                  s          2                θ            Adding the above two equations we get                              x          2                +        4                  y          2                                    =        4        (        c        o                  s          2                θ        +        s        i                  n          2                θ        )                                          =        4            Where we used the identity   c  o      s    2    θ  +  s  i      n    2    θ  =  1

Finding cartesian equation from parametric trigonometric equations I'm trying to find the cartesian equation of the curve which is defined parametrically by: x = 2 sin theta, y = cos^2 theta

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