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

ellipse main equatin is as follow:

X² / a² + Y² / b² = 1, where a≠0 and b≠0

for the first equation: x = 3 cos t and y = 8 sin t

we can write x² = 3² cos² t and y² = 8² sin² t

and then x² / 3² = cos² t and y²/8² = sin² t

therefore, x² / 3² + y²/8² = cos² t + sin² t = 1

equivalent to x² / 3² + y²/8² = 1

for the second equation, x = 3 cos 4t and y = 8 sin 4t we found

x² / 3² + y²/8² = cos² 4t + sin² 4t=1