Answer
The particle with position (x,y) moves 3 times clockwise along an ellipse.
The equation of this ellipse is: $\frac 1 {25}x^2 + \frac{1}{4}y^2= 1$
It starts and ends at $(0,-2)$
Work Step by Step
1. Convert the equation to a Cartesian one, so we can identify the type of curve:
$x = 5sin (t)$
$x^2 = 25sin^2(t)$
$\frac{1}{25}x^2 = sin^2(t)$
$y = 2cos(t)$
$y^2 = 4cos^2(t)$
$\frac{1}{4}y^2 = cos^2(t)$
Adding the equations:
$\frac 1 {25}x^2 + \frac{1}{4}y^2= sin^2(t) + cos^2(t)$
$\frac 1 {25}x^2 + \frac{1}{4}y^2= 1$
- As we can see by the equation, this curve is an ellipse with a radius of $\sqrt {25} = 5$ along the x-axis, and a radius of $\sqrt 4 = 2$ along the y-axis. And with its center at $(0,0)$
2. Find the initial and final point:
Initial point: $t = -\pi$
$x = 5 sin(-\pi) = 5(0) = 0$
$y = 2cos(-\pi) = 2(-1) = -2$
Final point: $t = 5 \pi$
$x = 5sin(5\pi) = 5(0) = 0$
$y = 2cos(5 \pi) = 2(-1) = -2$
- Since $sin(t)$ determines the $x$ position, and $cos(t)$ determines the $y$ position, the particle will move clockwise along the ellipse.
- For simple trigonometric functions like $sin(t)$ and $cos(t)$, the particle represented by $(x,y)$ completes a rotation after $t$ increasing by $2\pi$.
From $-\pi$ to $5\pi$, there is a total of $6\pi$.
Therefore, the particle does a total of 3 complete rotations along the elipse.
