37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


A circle centered at the origin with radius 4, generated counterclockwise

Verified step by step guidance

1. Recall the standard parametric equations for a circle centered at the origin with radius \( r \): \( x = r \cos(t) \) and \( y = r \sin(t) \).
2. Since the radius is 4, substitute \( r = 4 \) into the equations to get \( x = 4 \cos(t) \) and \( y = 4 \sin(t) \).
3. The parameter \( t \) represents the angle in radians measured from the positive x-axis, and it controls the position on the circle.
4. To generate the circle counterclockwise starting from the point \( (4,0) \), let \( t \) vary over the interval \( [0, 2\pi] \).
5. Thus, the parametric equations are \( x = 4 \cos(t) \), \( y = 4 \sin(t) \), with \( t \in [0, 2\pi] \).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.

Equation of a Circle

A circle centered at the origin with radius r satisfies x² + y² = r². To represent this circle parametrically, trigonometric functions sine and cosine are used, since they naturally describe circular motion.

Parameter Interval and Direction

The parameter interval defines the portion of the curve traced by the parametric equations. For a full circle traced counterclockwise, t typically ranges from 0 to 2π, with x = r cos t and y = r sin t ensuring the correct orientation.