Which of the following two sets of parametric functions both represent the same ellipse. Explain the difference between the graphs. x=3 cos t and y=8 sin t X=3 cos 4t and y = 8 sin 4t

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the answer: the main equation parametric of an  ellipse is  x²/a² + y²/b² = 1 a≠0 and b≠0 let's consider x=3 cos t and y=8 sin t, these are equivalent to x²=9 cos²t and y²=64 sin²t, and imiplying  x²/9=cos²t and y²/64=sin²t therefore, x²/9+y²/64= cos²t+ sin²t, but we know that cos²t+ sin²t =1 (trigonometric fundamental rule) so finally,  x²/9+y²/64=1 equivalent of  x²/3²+y²/8²= 1 this is an ellipse with the same method, we found x²/9+y²/64= cos²4t+ sin²4t =1, so the only difference between the graphs is the value of the angle (t and 4t)

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