Question

Give parametric equations that describe a circle of radius \(R,\) centered at the origin with clockwise orientation, where the parameter varies over the interval [0,10].

Short answer

Answer: The parametric equations are x(t) = R*cos(-t), 0 ≤ t ≤ 10 and y(t) = R*sin(-t), 0 ≤ t ≤ 10.

Step-by-step solution

Write down the standard parametric equations for a circle.

The standard parametric equations for a circle with radius R centered at the origin are given by: x = R*cos(t) y = R*sin(t)

Modify the equations for clockwise orientation.

To get a clockwise orientation, we need to use the negative of the standard parameter t in the equations. So the modified parametric equations are: x = R*cos(-t) y = R*sin(-t)

Set the interval for the parameter t.

The parameter t should vary over the interval [0, 10]. Therefore, we just need to indicate that t is in the interval [0,10]: 0 ≤ t ≤ 10

Write down the final parametric equations.

The final parametric equations of a circle with radius R, centered at the origin, with clockwise orientation, and parameter varying over the interval [0, 10] are: x(t) = R*cos(-t), 0 ≤ t ≤ 10 y(t) = R*sin(-t), 0 ≤ t ≤ 10

Key concepts

circle

A circle is a simple and beautifully symmetrical shape where all points are equidistant from a center point. This fixed distance is known as the radius. Circles are commonly represented in a two-dimensional plane using different equations, among which parametric equations offer a flexible way of describing a circle's path. Parametric equations are helpful in defining a circle as they allow the position of a point on the circle to be expressed as a function of a parameter, usually denoted as \( t \).

For a circle centered at the origin, the x and y coordinates of any point on the circle can be represented by:

• \( x = R \cdot \cos(t) \)
• \( y = R \cdot \sin(t) \)

where \( R \) is the radius of the circle and \( t \) is the parameter that varies.
By changing \( t \) across a specific interval, one can trace the entire circumference of the circle. This parametric representation is particularly useful in advanced mathematics and physics, where the dynamic nature of tracing a path is needed.

clockwise orientation

Clockwise orientation describes the direction in which points move along a path, like a circle, that follows the same direction as the hands of a clock. In mathematical terms when detailing a circle's path using parametric equations, this involves adjusting the usual equations to reflect this turned direction.

Typically, the orientation when tracing a circle using the parametric equations \( x = R \cdot \cos(t) \) and \( y = R \cdot \sin(t) \) is counterclockwise. To adjust these equations to a clockwise path, a crucial step is involving a negative angle change:

• \( x = R \cdot \cos(-t) \)
• \( y = R \cdot \sin(-t) \)

Here, substituting \( t \) with \( -t \) effectively reverses the typical counterclockwise tracing of the circle to a clockwise direction. This technique is effective as it flips the rotations, thanks to properties of trigonometric functions, giving us the desired orientation.

trigonometric functions

Trigonometric functions are a fundamental set of equations within mathematics that relate the sides and angles of triangles to each other. They become particularly powerful tools when analyzing and describing periodic phenomena, such as waves and circular motions. In the case of parametric equations for circles, the sine and cosine functions play pivotal roles.

These functions oscillate between -1 and 1, modeling circular motion effectively due to their periodic nature. In the context of a circle with radius \( R \), the trigonometric definitions applied are:

• Cosine function \( \cos(t) \): represents the horizontal position (x-coordinate) on the circle
• Sine function \( \sin(t) \): represents the vertical position (y-coordinate) on the circle

The angle \( t \) (often considered as time or another continuous parameter) sweeps out the path of the circle. By modifying \( t \), we adjust the movement along the path, with \( \cos(t) \) and \( \sin(t) \) ensuring the path maintains a perfect circular shape. Therefore, these trigonometric functions are essential in describing and solving many problems involving circular or oscillatory systems.