Problem 71 - Precalculus: A Concise Course

Question

With your graphing utility in radian and parametric modes, enter the equations \(\mathrm{X}_{1 \mathrm{~T}}=\cos \mathrm{T}\) and \(\mathrm{Y}_{1 \mathrm{~T}}=\sin \mathrm{T}\) and use the following settings. \(\operatorname{Tmin}=0, \operatorname{Tmax}=6.3,\) Tstep \(=0.1\) \(\operatorname{Xmin}=-1.5, \operatorname{Xmax}=1.5, \mathrm{Xscl}=1\) \(\operatorname{Ymin}=-1, \operatorname{Ymax}=1, \mathrm{Yscl}=1\) (a) Graph the entered equations and describe the graph. (b) Use the trace feature to move the cursor around the graph. What do the \(t\) -values represent? What do the \(x\) - and \(y\) -values represent? (c) What are the least and greatest values of \(x\) and \(y\) ?

Step-by-Step Solution

Answer

The graph represents a unit circle, where the \(t\)-values indicate the angle in radians, and the \(x\)- and \(y\)-values denote the cosine and sine values of that angle. The least value for both \(x\) and \(y\) is -1, while the greatest is 1.

Step 1: Plot The Parametric Equations

Enter the equations \(X_{1T} = \cos T\) and \(Y_{1T} = \sin T\), set the calculation mode to radian and parametric, and use the provided settings to create the plot. These settings include Tmin=0, Tmax=6.3, Tstep=0.1, Xmin=-1.5, Xmax=1.5, Xscl=1, Ymin=-1, Ymax=1, and Yscl=1.

Step 2: Graph Analysis and Utilization of the Trace Feature

The graph produced by the plotted equations represents a unit circle. By moving the cursor around the graph using the trace feature, it can be seen that the \(t\)-values represent the angle in radians, while the \(x\)- and \(y\)-values represent the corresponding cosine and sine values of that angle.

Step 3: Evaluate the Least and Greatest Values of \(x\) and \(y\)

Considering that the graph represents a unit circle, the least and greatest values of \(x\) and \(y\) relate to the boundaries of the circle. Hence, the minimum value for both \(x\) and \(y\) is -1 and the maximum value is 1.

Key Concepts

Understanding the Unit Circle

The unit circle is a foundational concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. This means any point on the circle has coordinates

• an x-coordinate given by \( \cos \theta \)
• a y-coordinate given by \( \sin \theta \)

These trigonometric functions, \( \cos \theta \) and \( \sin \theta \), correspond to the position along the x-axis and y-axis, respectively. The values of these functions cycle as we move around the circle. For each angle \( \theta \), these coordinates fall on the unit circle, illustrating how the cosine and sine functions relate directly to angles in circular motion.
Knowing this circle simplifies understanding of trigonometric functions and angles. On a graph, the unit circle helps visualize how these functions behave. This visualization aids in grasping complex trigonometric concepts, establishing a link between angles and real-world circular motion.

Trigonometric Functions and Graphing

Trigonometric functions like sine and cosine are essential in understanding periodic phenomena. Using parametric equations with these functions, such as \( X = \cos T \) and \( Y = \sin T \), allows graphing a full cycle of equivalent angles on a unit circle. When \( T \) represents the angle in radians, these parametric equations determine the x and y coordinates over a defined interval, such as from 0 to 6.3 radians.
Graphing these functions involves:

• Using radians to measure angles, ensuring the graph forms a trigonometric circle.
• Setting the graph limits (Xmin, Xmax, Ymin, Ymax) to capture the entire cycle of the unit circle.

Creating a graph using these points elucidates how angles produce repetitive patterns or cycles seen in natural phenomena and mechanical systems. Understanding trigonometric functions' graphical representation strengthens comprehension of their practical application.

Using a Graphing Utility

A graphing utility is a powerful tool for visualizing mathematical equations. With functions like \( X = \cos T \) and \( Y = \sin T \), the graphing utility assists in creating precise graphs of parametric equations representing the unit circle. To set up the utility for this task:

• Switch to radian mode, as this is the standard measure for angles in trigonometry.
• Set the parametric mode to plot the equations \( X = \cos T \) and \( Y = \sin T \).
• Input the Tmin, Tmax, and Tstep values for controlling the range and precision of the plotted graph.

These settings facilitate accurate plotting of the unit circle, enabling observation of the continuous nature of trigonometric functions. Using a graphing utility not only simplifies the plotting process but also enhances understanding of complex mathematical concepts through visual representation.

Exploring the Trace Feature

The trace feature in a graphing utility is extremely useful for examining detailed information about specific points on a graph. When you use this feature on the graph of a unit circle defined by the parametric equations \( X = \cos T \) and \( Y = \sin T \), each point cursor moves over reveals more about the circle's geometry.

• \( t \)-values indicate the angle in radians according to the graph's settings, representing the position along the unit circle.
• \( x \)-values show the corresponding cosine of the angle \( t \), demonstrating the circular progression along the x-axis.
• \( y \)-values reflect the sine of the angle \( t \), showing similar correspondence with the y-axis.

The feature aids in understanding the cyclic nature and relationship of the trigonometric functions as they relate to angles. Along with highlighting extremities, the trace feature's insight into least and greatest \( x \)- and \( y \)-values (both being -1 and 1) further reinforces the symmetry inherent in trigonometric circles.