Final answer:
The parametric equations for a circle of radius 2 centered at the origin and oriented clockwise are x(t) = 2 cos(t) and y(t) = -2 sin(t), with the negative sign in the y equation indicating the clockwise direction.
Step-by-step explanation:
To find the parametric equations for a circle of radius 2 centered at the origin, oriented clockwise, we start by recalling the standard equations for a circle moving in a counter-clockwise direction:
- x(t) = r cos(t)
- y(t) = r sin(t)
Where r is the radius of the circle, and t is the parameter, typically representing time. Since our circle is oriented clockwise, we can modify these equations to reverse the direction:
- x(t) = 2 cos(-t)
- y(t) = 2 sin(-t)
By the trigonometric identities for cosine and sine, which state that cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), the equations can be further simplified to:
- x(t) = 2 cos(t)
- y(t) = -2 sin(t)
Therefore, the parametric equations for the give circle are:
- x(t) = 2 cos(t)
- y(t) = -2 sin(t)
The negative sign in the equation for y(t) ensures that the motion is in the clockwise direction.