Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) for all real numbers \(x\) in the interval \([-2 \pi, 2 \pi]\) Use the graphs to find the real numbers \(x\) such that \(y_{1}=y_{2}.\) $$\begin{aligned} &y_{1}=\sin x\\\ &y_{2}=-\frac{1}{2} \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are approximately at \(x = -4.712, -1.571, 1.571, 4.712\) (the precise values differ slightly due to decimal approximations). These are the points where the curve \(y_{1} = \sin x\) intersects with the line \(y_{2} = -\frac{1}{2}\) in the interval \([-2 \pi, 2 \pi]\).

Step 1: Graphing \(y_{1} = \sin x\)

Use a graphing utility to graph \(y_{1}=\sin x\) in the interval \([-2\pi, 2\pi]\). The sine function has a period of \(2\pi\), and within one period, it ranges from -1 to 1. Since \(\sin x\) waves between -1 and 1, it crosses the x-axis at values of ¬\(-(2n - 1)\frac{\pi}{2}\), where n is an integer

Step 2: Graphing \(y_{2}= -\frac{1}{2}\)

Now, graph the function \(y_{2}=-\frac{1}{2}\). This is a constant function and its graph is a horizontal line across \(-\frac{1}{2}\) in the y-axis.

Step 3: Find the points of intersection

Find the points of intersection of the two graphs as they represent the values of \(x\) for which \(y_{1} = y_{2}\). This is done by examining where the graph of the sine wave intersects with the constant function at \(-\frac{1}{2}\).

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions, commonly denoted as \(y = \sin x\). It describes a smooth, periodic wave that oscillates above and below the x-axis. For any given angle \(x\), the sine function returns the y-coordinate of the corresponding point on the unit circle. The range of the sine function is between -1 and 1, capturing the height of the wave.
In graphing \(y = \sin x\), you will notice that it creates a repeating wave pattern, called a sinusoidal wave. This wave reaches its peak at +1 and its trough at -1. At the points where it crosses the x-axis, called zeros, the function value is 0. These x-intercepts occur at multiples of \(\pi\), specifically at \(x = n\pi\), where \(n\) is an integer. This periodicity is a key characteristic that defines the behavior of the sine function.

Period of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function \(y = \sin x\), the period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the pattern of the wave repeats itself. Understanding the period helps in predicting the wave behavior across different values of \(x\).
The period is crucial when adjusting the sine function for transformations, such as stretching or compressing the wave. By manipulating coefficients, you can change the period, resulting in frequency changes of the wave. Still, the standard sine function always returns to its original shape after \(2\pi\).

• Recognize that other trigonometric functions might have different periods.
• This feature allows for modeling periodic phenomena like sound waves or tides.

Graphing Utility

A graphing utility is a tool or software used to visualize mathematical functions graphically. With a graphing utility, you can input various functions and instantly see the resulting curves. This tool is incredibly helpful for understanding complex functions like the sine function, especially in identifying behavior, such as its periodic nature.
Graphing utilities often include features for adjusting the scale or the range of the graph displayed. You can focus on particular intervals, such as \([-2\pi, 2\pi]\), allowing you to closely observe the interactions between different functions plotted.

• Use graphing utilities to easily overlay multiple functions, facilitating the comparison between them.
• You'll also find features for finding precise intersection points or important characteristics of the functions, such as minima and maxima.

Intersection of Graphs

Intersection of graphs involves finding points where two or more functions meet or cross each other on a graph. In the context of the exercise, this would involve determining where \(y_1 = \sin x\) and \(y_2 = -\frac{1}{2}\) intersect. These points of intersection represent the x-values where both functions have the same value.
For the given exercise, you'll notice that this involves solving the equation \(\sin x = -\frac{1}{2}\). The intersection points are the solutions that can be seen graphically where the sine wave meets the horizontal line \(y = -\frac{1}{2}\).

• Identifying these intersections can help in solving equations graphically.
• These intersections may require the use of graphing utilities or algebra to find exact solutions, especially if the functions involved are complex.

Understanding how and where graphs intersect is crucial for many applications in math, from calculus to real-world problem-solving like engineering or physics.