Problem 91 - Algebra and Trigonometry

Question

The plane curve described by the parametric equations \(x=3 \cos t \quad\) and \(\quad y=3 \sin t, \quad 0 \leq t<2 \pi, \quad\) has \(\quad \mathrm{a}\) counterclockwise orientation. Alter one or both parametric equations so that you obtain the same plane curve with the opposite orientation.

Step-by-Step Solution

Answer

The parametric equations for the same curve but with reversed orientation are \(x = 3 \cos t\) and \(y = -3 \sin t\).

Step 1: Analysing and understanding the parametric equations

Analyse and understand the given parametric equations \(x=3 \cos t\) and \(y=3 \sin t\), which represent a circle of radius 3. The circle is traversed in a counterclockwise direction as t increases from 0 to \(2\pi\).

Step 2: Inverting the orientation

Invert the orientation by negating the parameter of the trigonometric functions in one of the equations. If we negate the parameter in the second equation, we obtain \(x=3 \cos t\) and \(y=3 \sin -t\). But since \(\sin -t = - \sin t\), the new set of equations will be \(x=3 \cos t\) and \(y=-3 \sin t\). These still represent a circle of radius 3, however, now it will be traversed in the clockwise direction, meaning the orientation of the curve is inverted.

Key Concepts

Trigonometric Parametric Equations

Understanding trigonometric parametric equations involves recognizing that these equations express the coordinates of points that make up a curve using trigonometric functions of one parameter, often denoted as t. Consider a straightforward example: the equations x = r cos(t) and y = r sin(t), where r is a constant and t varies within a certain interval.

These parametric equations describe a circle with radius r in the Cartesian plane, with (x, y) being any point on the circle and t corresponding to the angle in radians from the positive x-axis to the radius that passes through (x, y). The beauty of parametric equations lies in their ability to elegantly represent complex curves, such as circles, ellipses, and more intricate shapes, by tying every point on a curve to a single parameter.

For the exercise provided, the parameter t ranges from 0 to 2π, describing a full revolution around the circle. As the value of t increases within this range, the point (x, y) follows the path of the circle in a counterclockwise direction.

Circle Parametric Representation

Circles are one of the basic geometric shapes that can be elegantly described using parametric equations. A circle's parametric representation enables us to encode the x and y coordinates of any point on the circle's circumference as functions of a single variable, usually an angle. For a Given circle of radius r, the parametric equations are formulated as x = r cos(θ) and y = r sin(θ), where θ represents the angle parameter.

These equations create a set of points that trace the circle as θ varies from 0 to 2π. This interval completes a full revolution around the circle, hence the curve described by these parametric equations will start and end at the point (r, 0) on the Cartesian plane. By plugging in specific values of θ, you can determine the exact position on the circle's perimeter for that angle, which is particularly useful in applications ranging from computer graphics to engineering design.

Inverting Curve Orientation

The orientation of a curve refers to the direction in which it is traversed as the parameter increases. In the context of parametric equations, particularly those describing circles, orientation can be either clockwise or counterclockwise. In the original exercise, the circle is traversed counterclockwise, as indicated by the increasing parameter t.

To invert this curve's orientation, one must manipulate the equations to reverse the traversal direction. This is achieved by shifting what happens to the angle parameter in the trigonometric functions. For a circle: swapping sin(t) to sin(-t) or cos(t) to cos(-t) effectively reverses the direction because the trigonometric functions are odd (for sine) or even (for cosine) functions. Therefore, a negative angle traverses the circle in the opposite direction. For the given problem, altering the sign of t in the y component leads to the curve being traversed clockwise. This effectively inverts the orientation while maintaining the shape and position of the curve, allowing for versatile manipulation of the parametric representation.