Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Describe the graph of a curve \(C\) that has the parametrization $$ x=3+2 \sin t, \quad y=-2+2 \cos t ; \quad 0 \leq t \leq 2 \pi . $$ (b) Change the parametrization to \(x=3-2 \sin t, \quad y=-2+2 \cos t, \quad 0 \leq t \leq 2 \pi\) and describe how this changes the graph from part (a). (c) Change the parametrization to \(x=3-2 \sin t, \quad y=-2-2 \cos t ; \quad 0 \leq t \leq 2 \pi\) and describe how this changes the graph from part (a).

Step-by-Step Solution

Answer

(a) Circle at (3,-2) radius 2, counterclockwise. (b) Same circle, clockwise. (c) Same circle, vertical flip, clockwise.

Step 1: Identify the Parametrization in Part (a)

For the first given parametrization, the equations are \(x=3+2 \sin t\) and \(y=-2+2 \cos t\) for \(0 \leq t \leq 2\pi\). These equations represent a circle centered around the point \((3, -2)\) with radius 2.

Step 2: Understand the Circle in Part (a)

The standard form of a parametric circle is \((x, y) = (h + r \sin t, k + r \cos t)\). Here, \(h = 3\), \(k = -2\), and \(r = 2\). Thus, the graph is a circle centered at \((3, -2)\) with radius 2, traversed counterclockwise from \((3, 0)\) starting when \(t = 0\).

Step 3: Change the Parametrization in Part (b)

The new parametrization is \(x = 3 - 2 \sin t\) and \(y = -2 + 2 \cos t\). This means the center remains the same, but the traversal direction changes to clockwise for the circle centered at \((3, -2)\), as reversing the sine sign effectively mirrors the path.

Step 4: Describe Changes in Part (b)

The graph still describes a circle of radius 2 centered at \((3, -2)\), but the path around this circle is now clockwise. This is due to the reversed sine term, which affects the direction of traversal without altering the shape.

Step 5: Change the Parametrization in Part (c)

The parametrization changes to \(x = 3 - 2 \sin t\), \(y = -2 - 2 \cos t\). The effect of changing the cosine term is a vertical flip of the circle, making the new center point reflection across the original horizontal axis.

Step 6: Describe Changes in Part (c)

The graph remains a circle of radius 2, but it is centered at \((3, -2)\) with its original position inverted (vertical reflection) compared to the initial configuration in part (a). The traversal direction remains clockwise as in part (b), because the flip in cosine doesn't change direction inherent in traversal.

Key Concepts

Circle Graphs

A circle graph, often known as a parametric or trigonometric circle, depicts the path traced by a point moving in a circular trajectory. In parametric equations, both the x and y coordinates are defined explicitly by separate functions of a parameter, commonly denoted by \(t\) in the unit interval \([0,2\pi]\). These graphs embody a seamless way to express complex circular motion that is challenging to capture with standard Cartesian equations.

For circles, the parametric equations are typically presented in the form \((x, y) = (h + r \sin t, k + r \cos t)\), where \(h\) and \(k\) denote the horizontal and vertical shifts from the origin, representing the center, while \(r\) signifies the radius of the circle. This provides an efficient framework for generating a circle's graphical representation, which is easily adjustable by changing the parameter values.

Center and Radius

The center and radius of a circle in a parametric equation significantly determine the circle's position and size in a coordinate plane. Understanding these parameters helps you visualize and manipulate circle graphs effectively.

The center of a circle \((h, k)\) is essentially the pivot about which the circular path rotates. For example, the equation \((x, y) = (3 + 2 \sin t, -2 + 2 \cos t)\) has its center at \((3, -2)\). This means our circle is shifted three units to the right and two units down relative to the origin.

The radius \(r\) in the equation gives the distance from the center to any point on the circle. In our parametric equations, this is represented by the coefficients before \(\sin t\) and \(\cos t\), which are both 2 in our examples. This value tells us that we've got a circle with a radius of 2, maintaining its standard circular proportion based on how the sine and cosine functions operate, radiating uniformly from the center.

Traversal Direction

Traversal direction describes how a point moves around a circle graph when a parameter varies. In our given scenarios, this direction is dictated by how the functions of \(\sin t\) and \(\cos t\) are configured.

The parametric equation \((x, y) = (3 + 2 \sin t, -2 + 2 \cos t)\) starts at point \((3, 0)\) and moves counterclockwise, the default direction when \(t\) increases from 0 to \(2\pi\). This conventionary motion is reversed when you alter the sine term from positive to negative, leading to a mirrored traversal direction.

For example, with the equation \((x, y) = (3 - 2 \sin t, -2 + 2 \cos t)\), the traversal changes from counterclockwise to clockwise while still covering the circumference. This change is essential in understanding how function alterations in parametric equations can modify the directional path without impacting the shape or size of the circle.

Parametrization Changes

Parametrization changes occur when we modify the equations governing our circle's path, impacting the circle's orientation or direction. These manipulations include changing coefficients, altering signs, or swapping functions, resulting in marked differences in the graph's traversal or appearance.

In our exercises, adjustments in the sine or cosine terms triggered changes. For instance, shifting from \(\sin t\) to \(-\sin t\) altered how the circle was traversed, yet maintained the original shape and size. Similarly, inversely flipping \(\cos t\) in the equation from positive to negative resulted in a vertical flip, a reflection over the horizontal center line.

These deliberate changes don't alter the fundamental properties of the graph, like radius or center point, but provide dynamic ways for expressing and visualizing circular motion across different scenarios. Hence, mastering these principles can help you simulate a myriad of circular paths by fine-tuning equation parameters expertly.