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. Step 1: Recognize that the given rectangular equation x² + y² = 25 represents a circle centered at the origin with a radius of 5. Parametric equations are often used to describe such curves in terms of a parameter, typically t.
2. Step 2: Recall the standard parametric equations for a circle: x = r * cos(t) and y = r * sin(t), where r is the radius of the circle and t is the parameter (angle in radians).
3. Step 3: Substitute the radius of the circle, r = 5, into the parametric equations. This gives x = 5 * cos(t) and y = 5 * sin(t).
4. Step 4: Verify that these parametric equations satisfy the original rectangular equation. Substitute x = 5 * cos(t) and y = 5 * sin(t) into x² + y² = 25. Using trigonometric identities, cos²(t) + sin²(t) = 1, the equation holds true.
5. Step 5: Conclude that the parametric equations x = 5 * cos(t) and y = 5 * sin(t) correctly describe the circle given by the rectangular equation x² + y² = 25.