Answer:
E(t) = [ 4cos(t), 2 sin(t) ]
Step-by-step explanation:
The equation for the ellipse can be represented as:
(x² ÷ a² ) + (y² ÷ b² ) = 1 Here, a and b symbolize the semi-major and semi-minor axes.
In this specific instance, we have x²/16 + y²/ 4 = 1
This expression indicates that it follows the form of (x/a)² + (y/b)² =1
Specifically in our case, x²/16 + y²/4 = 1, which resembles sin²α + cos²α = 1.
By setting x = 4 cos(t), we can proceed to calculate y.
Utilizing the equation x²/16 + y²/4 = 1, we find that:
x²/16 + y²/4 = 1 ⇒ (x² + 4y²) ÷ 16 = 1
Solving for y yields: (x² + 4y²) = 16 ⇒ y² = ( 16 - x²) ÷4
Substituting x = 4 cos(t) gives us y² = (16 - 16cos²(t)) ÷ 4, leading to y = √4 (1 - cos²(t)).
Consequently, we have y = 2 sin(t).
Thus, the vector parameterization of the ellipse can be stated as:
E(t) = [ 4cos(t), 2 sin(t) ]
