Write parametric equations for the rectangular equation below.
$x^2+y^2=25$

• A. $x=25\sin t$; $y=25\cos t$
• B. $x=25\cos t$; $y=25\sin t$
• C. $x=5\sin t$; $y=5\cos t$
• D. $x=5\cos t$; $y=5\sin t$

Answer: (D)

Verified step by step guidance

1. Identify the given rectangular equation: x^2 + y^2 = 25. This represents a circle centered at the origin with a radius of 5.
2. Recall the parametric form of a circle: x = r * cos(t) and y = r * sin(t), where r is the radius and t is the parameter (angle in radians).
3. Since the radius of the circle is 5, substitute r = 5 into the parametric equations: x = 5 * cos(t) and y = 5 * sin(t).
4. Verify that these parametric equations satisfy the original rectangular equation by substituting x and y back into x^2 + y^2 = 25.
5. Conclude that the correct parametric equations are x = 5 * cos(t) and y = 5 * sin(t), which describe the same circle as the given rectangular equation.