Question: Parameterize the circle using: $x=2\cos t$, $y=2\sin t$, $t \in [0,2\pi]$ what is this?
1. **Problem Statement:** We are given the parametric equations $x=2\cos t$ and $y=2\sin t$ with the parameter $t$ ranging from $0$ to $2\pi$. We want to understand what these equations represent.
2. **Formula and Explanation:** The parametric equations $x=a\cos t$ and $y=a\sin t$ describe a circle centered at the origin with radius $a$. Here, $a=2$.
3. **Intermediate Work:**
- The equation of a circle centered at the origin with radius $r$ is $x^2 + y^2 = r^2$.
- Substitute $x=2\cos t$ and $y=2\sin t$:
$$x^2 + y^2 = (2\cos t)^2 + (2\sin t)^2 = 4\cos^2 t + 4\sin^2 t = 4(\cos^2 t + \sin^2 t)$$
- Using the Pythagorean identity $\cos^2 t + \sin^2 t = 1$, we get:
$$x^2 + y^2 = 4 \times 1 = 4$$
4. **Conclusion:** The parametric equations describe a circle centered at the origin with radius $2$. The parameter $t$ represents the angle in radians, sweeping from $0$ to $2\pi$ to trace the entire circle.
**Final answer:** The given parametric equations represent a circle of radius $2$ centered at the origin.