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

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)