Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x$$

Step-by-Step Solution

Answer

After graphing and filling in the table, it becomes clear that when \(x\) is between 0 and \(\frac{\pi}{2}\) (which is the \(x\) range in the exercise), the two functions \(y_{1}=\cos \left(\frac{\pi}{2}-x\right)\) and \(y_{2}=\sin x\) are actually equivalent and yield the same \(y\) values.

Step 1: Input Functions Into Graphing Utility

Open your graphing utility and input both functions \(y_{1}=\cos \left(\frac{\pi}{2}-x\right)\) and \(y_{2}=\sin x\). You should see two wave-patterned lines cross your graph.

Step 2: Analyze the Graphs

Notice that \(y_{1}\) and \(y_{2}\) have the same wave-pattern, shifting. The cosine function \(y_{1}\) shifts to the right compared to the sine function \(y_{2}\). Because of the nature of these two functions, they will overlap every \(\frac{\pi}{2}\). We can use this information to conjecture about the relationship between the two values of \(y_{1}\) and \(y_{2}\) at different \(x\) values.

Step 3: Use the Graph to Fill in the Table

Take the given \(x\) values from the table (0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4) and calculate the corresponding \(y_{1}\) and \(y_{2}\) values using the plotted functions. Fill these values into the table.

Step 4: Observe Patterns and Formulate a Conjecture

After calculating and filling in the \(y_{1}\) and \(y_{2}\) values, observe any patterns across the x-values. Formulate a conjecture about \(y_{1}\) and \(y_{2}\). Due to the transformation of the cosine function, when \(y_{1}=\cos \left(\frac{\pi}{2}-x\right)\), it is equivalent to \(y_{2}=\sin x\). Hence we can conjecture \(y_{1}=y_{2}\) for values of \(x\).

Key Concepts

Using Graphing Calculators

In today's technologically advanced classrooms, graphing calculators have become essential tools for understanding complex mathematical concepts, especially in trigonometry. A graphing calculator allows students to visualize functions easily, which is particularly helpful when it comes to graphing trigonometric functions. To use a graphing calculator for trig functions:

• First, input the trigonometric function into the calculator.
• Adjust the viewing window to ensure the important features of the function are visible.
• Use the calculator's graphing capability to observe the shape and pattern of the function.
• Finally, analyze the graph to understand the behavior of the trigonometric function over different intervals.

Graphing calculators not only help in plotting the functions but also in exploring properties such as periodicity and symmetry, making conjectures, and understanding transformations.

Trigonometry Concepts

Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles, particularly right-angled triangles. Its fundamental functions include sine, cosine, and tangent, which are essential for solving problems involving angles and dimensions.

Understanding the Unit Circle

The unit circle is a powerful concept in trigonometry used to define the trigonometric functions for all angles. It helps relate angles to the coordinates of a point on a circle of radius one, thus linking angular measures to the lengths of line segments.

Angles and Their Measurement

Angles in trigonometry are typically measured in radians or degrees, with radians being the more mathematically natural unit as it relates directly to the arc length on the unit circle.

Sine and Cosine Relationship

The sine and cosine functions are closely related trigonometric functions that represent periodic oscillations. They have a special relationship expressed by several identities, the most noteworthy being the co-function identity:
\[\begin{equation}\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)\end{equation}\]This identity indicates that the sine of an angle is equal to the cosine of its complement. This relationship explains why the graphs of sine and cosine functions are similar; they are essentially the same waves shifted horizontally by \begin{math}\frac{\pi}{2}\begin{math} radians. This property is evident when plotting both functions on a graphing calculator, as their similarity and transformation can be visually compared.

Function Transformation

In mathematics, function transformation involves altering a parent function's graph to produce a new function. There are several types of transformations, including translations, reflections, stretches, and compressions. Essential transformations for trigonometric functions involve:

• Translations: Shift the graph horizontally or vertically without changing its shape.
• Reflections: Flip the graph across a specified axis, altering the graph's orientation.
• Stretches and Compressions: Change the amplitude or period of the wave without affecting its basic shape.

Understanding how to apply these transformations allows students to graph more complex trigonometric functions and recognize patterns in their equations and graphs, aiding in the prediction of their behavior.