Questions: Question 18 Describe the plane curve represented by the parametric equations by eliminating the parameter, x=4 sin t and y=4 cos t ; 0 ≤ t ≤ 2 π Ellipse centered at (0,0) and vertices at (4,0) and (-4,0) Circle centered at (0,0) and radius 2 Circle centered at (4,4) and radius 16 Circle centered at (0,0) and radius 4

Solution Steps

The parametric equations given are: \[ x = 4 \sin t \] \[ y = 4 \cos t \] for \( 0 \leq t \leq 2\pi \).

We square both equations to eliminate the parameter \( t \): \[ x^2 = (4 \sin t)^2 = 16 \sin^2 t \] \[ y^2 = (4 \cos t)^2 = 16 \cos^2 t \]

Next, we add the squared equations: \[ x^2 + y^2 = 16 \sin^2 t + 16 \cos^2 t \]

Using the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \), we can simplify the equation: \[ x^2 + y^2 = 16(\sin^2 t + \cos^2 t) = 16 \cdot 1 = 16 \]

Thus, the equation representing the plane curve is: \[ x^2 + y^2 = 16 \] This indicates a circle centered at the origin \((0,0)\) with a radius of \(4\).

Final Answer