Question

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. $$\text { Involute of a circle } x=\cos t+t \sin t, y=\sin t-t \cos t$$

Short answer

Answer: The appropriate interval for the parameter "t" is from \(0\) to \(2\pi\).

Step-by-step solution

Understanding the involute of a circle

The involute of a circle is the curve that is traced by a point, which is located on the end of a line that is tangent to the circle and starts to unroll or move in a straight line. In this case, the given curve is represented by the parametric equations: $$ x(t) = \cos{t} + t\sin{t}, \quad y(t) = \sin{t} - t\cos{t}. $$

Find the interval of parameter t

In order to determine the range for the parameter "t", we need to find the time at which the moving point moves around the entire circle and begins repeating its path. Since the point is moving along the circle, one complete revolution around the circle will cover all the features of involute. As we are dealing with a circle, we know that it has a circumference of \(2\pi r\). For a unit circle, the radius \(r=1\), so the entire circumference is \(2\pi\). Therefore, letting "t" range between \(0\) and \(2\pi\) covers the whole curve.

Graph the curve using a graphing utility

Now that we have determined the range of the parameter "t", we can graph the curve using a graphing utility of our choice. There are various tools available, such as Desmos or GeoGebra, that will allow you to input the parametric equations of the involute and specify the range for the parameter "t".

Input the equations and set the parameter range

In the graphing utility, input the following parametric equations: $$ x(t) = \cos{t} + t\sin{t}, \quad y(t) = \sin{t} - t\cos{t}. $$ Set the range for the parameter "t" from \(0\) to \(2\pi\).

Observe the graph

Once the graphing utility plots the involute of the circle, analyze the graph and note its features. You should observe a continuous curve without any abrupt stops or gaps. By choosing the correct interval for the parameter "t", we have been able to graph the entire involute of the circle, capturing all its features of interest.

Key concepts

Involute of a Circle

The involute of a circle is a fascinating mathematical concept that involves tracing a curve from a circle. Imagine holding a string attached to the end of a circle. As you unroll the string while keeping it taut, the path it follows forms what is known as an involute. The involute is described by parametric equations. For a circle, these equations are: \( x(t) = \cos{t} + t\sin{t} \) for the x-coordinate, and \( y(t) = \sin{t} - t\cos{t} \) for the y-coordinate. These equations allow us to calculate the x and y positions of the curve at any given point along the path as t varies. Understanding how these equations work can help in visualizing complex shapes and paths that occur in mechanical gears and clocks.

Graphing Utilities

Graphing utilities are immensely valuable when it comes to plotting complex equations like the involute of a circle. These tools simplify the graphing process by allowing you to input equations and observe their plotted shapes instantly. Popular graphing utilities like Desmos and GeoGebra make it easy to graph using parametric equations. Simply input your equations:

• \(x(t) = \cos{t} + t\sin{t}\)
• \(y(t) = \sin{t} - t\cos{t}\)

Graphing software understands these and draws the shapes so you can visualize what might be difficult to imagine otherwise. This visual aid is particularly useful in educational contexts and for exploring different mathematical concepts.

Parametric Curve Graphing

Parametric curve graphing is a method used to trace curves on a plane using equations that express the coordinates as functions of a single parameter. In mathematics, parametric equations describe complex shapes easily and offer flexibility in graphing. Using the equations \(x(t) = \cos{t} + t\sin{t}\) and \(y(t) = \sin{t} - t\cos{t}\), we can examine how the curve forms. Instead of plotting x directly against y, parametric graphing ties these variables to a common parameter, t. This method is particularly useful for representing curves that overlap or have a looping structure, precisely like an involute of a circle. The parametric approach allows us to see how t influences both x and y simultaneously, making it easier to understand the curve's full path.

Interval for Parameter t

Determining the correct interval for the parameter \(t\) is crucial to capturing the entire curve of interest. For an involute of a circle, which involves wrapping and unwrapping, it's important to choose a suitable range for \(t\) to see one full revolution. Since the curve unrolls from the circle, \(t\) often changes from 0 to some multiple of \(2\pi\) (the circle's circumference for a unit circle). For a full representation of the involute, setting \(t\) between \(0\) and \(2\pi\) ensures that all the features are included. Sticking to this range guarantees that the entire curve is drawn without missing any parts, while also preventing repetition. It's an elegant way to ensure the complete exploration of the curve's geometry.