Problem 28 - Precalculus with Calculus Previews

Question

Graphically show the difference between the given curves. Assume that \(a>0\) and \(b>\) o, $$ \begin{array}{l} x=a \cos \frac{t}{2}, y=a \sin \frac{t}{2}, \quad 0 \leq t \leq \pi \\ x=a \cos \left(-\frac{t}{2}\right), y=a \sin \left(-\frac{t}{2}\right), \quad-\pi \leq t \leq 0 \end{array} $$

Step-by-Step Solution

Answer

The curves represent the top and bottom halves of a circle when combined.

Step 1: Understand the Parametric Equations

We are given two sets of parametric equations. The first set is \( x = a \cos\left(\frac{t}{2}\right) \) and \( y = a \sin\left(\frac{t}{2}\right) \) with \( 0 \leq t \leq \pi \). The second set is \( x = a \cos\left(-\frac{t}{2}\right) \) and \( y = a \sin\left(-\frac{t}{2}\right) \) with \( -\pi \leq t \leq 0 \). Both equations describe arcs of circles using parameter \( t \).

Step 2: Simplify the Parametric Equations

Since \(\cos\) and \(\sin\) are even and odd functions respectively, we have:1. \(\cos\left(-\frac{t}{2}\right) = \cos\left(\frac{t}{2}\right)\). 2. \(\sin\left(-\frac{t}{2}\right) = -\sin\left(\frac{t}{2}\right)\).This implies the second set of equations represents a reflection across the x-axis of the curve represented by the first set.

Step 3: Identify the Shapes of the Curves

Both parametric equations describe semi-circular arcs. For the first equation, as \(t\) varies from 0 to \(\pi\), it forms the upper half of a circle of radius \(a\). For the second equation, as \(t\) varies from \(-\pi\) to 0, it forms the lower half of the same circle (since the x-coordinate is unchanged but the y-coordinate is reflected).

Step 4: Graph the Curves

To graph both sets of parametric equations, consider a coordinate system where:- The first curve, \( (x, y) = (a \cos\frac{t}{2}, a \sin\frac{t}{2}) \), traces out the upper semi-circle from \((a, 0)\) to \((-a, 0)\).- The second curve, \( (x, y) = (a \cos\frac{t}{2}, -a \sin\frac{t}{2}) \), traces out the lower semi-circle from \((a, 0)\) to \((-a, 0)\).

Step 5: Visualize the Entire Circle

In conjunction, these curves form a complete circle centered at the origin with radius \(a\). The first set forms the top half from \(x = a\) to \(x = -a\) above the x-axis, and the second set forms the mirrored bottom half of the circle below the x-axis.

Key Concepts

Circular Arcs

When we talk about circular arcs drawn using parametric equations, we are discussing segments of a circle, described step by step, as the parameter varies. Here's how it works:

• The circle is positioned in a plane with a fixed radius.
• Using trigonometric functions, we can specify the position along the circle via angle parameters. For example, \(x = a \cos\left(\frac{t}{2}\right)\) and \(y = a \sin\left(\frac{t}{2}\right)\) tell us the x and y positions on the circle based on angle \(t\).
• In the example, as \(t\) moves from \(0\) to \(\pi\), they trace out the top half of the circle, starting and ending at the x-axis.

Understanding these arcs as parts of a circle helps in visualizing their layout. For our equations, the arc of the circle is determined by \(t\), subtly moving our position along the circle's curve. By visualizing this on a graph, we see how these parameters come together to create the shape we anticipate, forming half a circle when treated within their specified bounds.

Graphing Curves

Graphing parametric curves involves plotting points that follow a set of rules to showcase a specific shape. This concept is central for grasping how curves are visualized on a graph. For the circular arcs we've discussed:

• Position the center of your graph at the origin, \( (0,0)\).
• Calculate points incrementally. For instance, with the first equation \(x = a \cos\left(\frac{t}{2}\right)\) and \(y = a \sin\left(\frac{t}{2}\right)\), you plot points where \(0 \leq t \leq \pi\).
• These plots depict, point-by-point, one smooth half of a circle.

For graphing, it's crucial to understand how each coordinate pair gives a point on the curve. Align each result with the parameter it represents. Graphing these points results in understanding the entirety of the curve's layout—critical in mathematical and visualization tasks. Every step along \(t\) is another step in building the circle's shape.

Reflection Across the X-axis

Reflection across the x-axis can be thought of as a mirror image of a curve. This concept is evident in our parametric equations. When you follow these steps, you can see the result:

• In the context of parametric equations, modifying the equation for the y-coordinate, such as using negative signs, reflects the curve across the x-axis.
• The second set, \( x = a \cos\left(-\frac{t}{2}\right) \) and \( y = a \sin\left(-\frac{t}{2}\right) \), switches the y direction but keeps x the same. This reflects the curve downwards.

Understanding these reflections can help in seeing symmetry within graphs. It can simplify many scenarios by directly translating the effects of parameter changes into visible, intuitive patterns. In our exercise, both upper and lower halves of the circle illustrate reflecting across the x-axis perfectly, showing the entire circle as a unified result.