Which of the following parametric equations represents a circle of radius $3$ centered at the origin, traced counterclockwise as $t$ increases from $0$ to $2\pi$?

• A. $x=3\cos \left(t\right)$, $y=3\sin \left(t\right)$
• B. $x=3\sin \left(t\right)$, $y=3\cos \left(t\right)$
• C. $x=3\sin \left(t\right)$, $y=3\sin \left(t\right)$
• D. $x=3\cos \left(t\right)$, $y=3\cos \left(t\right)$

Answer: (A)

Verified step by step guidance

1. Recall that the parametric equations for a circle of radius \( r \) centered at the origin are given by \( x = r \cos(t) \) and \( y = r \sin(t) \), where \( t \) is the parameter representing the angle in radians.
2. Since the radius is 3, substitute \( r = 3 \) into the general parametric form to get \( x = 3 \cos(t) \) and \( y = 3 \sin(t) \).
3. The parameter \( t \) should vary from 0 to \( 2\pi \) to trace the entire circle once counterclockwise, because \( \cos(t) \) and \( \sin(t) \) complete one full cycle over this interval.
4. Check the direction of tracing: as \( t \) increases from 0 to \( 2\pi \), \( (x,y) = (3\cos(t), 3\sin(t)) \) moves counterclockwise around the circle, which matches the problem's requirement.
5. Verify that the other options do not satisfy the conditions for a circle of radius 3 centered at the origin traced counterclockwise by examining their forms and the behavior of \( x \) and \( y \) as functions of \( t \).