Problem 79 - Precalculus: Functions and Graphs

Question

Use a graph to solve the inequality on the interval \([-\pi, \pi]\) $$\cos (2 x-1)+\sin 3 x \geq \sin \frac{1}{3} x+\cos x$$

Step-by-Step Solution

Answer

Graph both functions, find intersection points, and test the intervals to determine where the inequality holds.

Step 1: Understand the Inequality

The given inequality is \( \cos(2x - 1) + \sin(3x) \geq \sin\left(\frac{1}{3}x\right) + \cos(x) \). Our task is to find the values of \( x \) in the interval \([-\pi, \pi]\) where this inequality holds true.

Step 2: Graph the Functions

Graph the functions \( f(x) = \cos(2x - 1) + \sin(3x) \) and \( g(x) = \sin\left(\frac{1}{3}x\right) + \cos(x) \) on the same set of axes for the interval \([-\pi, \pi]\). This will allow us to visually compare which function is greater.

Step 3: Identify Intersection Points

Observe the points where the two graphs intersect. These points will help in determining the sections of \( x \) where the inequality changes direction. Solve \( \cos(2x - 1) + \sin(3x) = \sin\left(\frac{1}{3}x\right) + \cos(x) \) to find these exact intersection points.

Step 4: Test Intervals Between Intersection Points

Test the inequality in intervals defined by the intersection points found in Step 3 to determine where \( f(x) \geq g(x) \). At each point within an interval, if \( f(x) \geq g(x) \), then the inequality holds true for the entire interval.

Step 5: Combine Results

Combine the intervals from Step 4 where \( f(x) \geq g(x) \) to obtain the solution set for \( x \) in \([-\pi, \pi]\) where the inequality is satisfied.

Key Concepts

Graphical solutions

Graphical solutions are a powerful tool in solving trigonometric inequalities. They allow us to visualize the functions over a specific domain and identify where one function is greater than or equal to another. For the given inequality, the strategy involves plotting two trigonometric functions:

• The first function is \( f(x) = \cos(2x - 1) + \sin(3x) \).
• The second function is \( g(x) = \sin\left(\frac{1}{3}x\right) + \cos(x) \).

Both functions need to be plotted over the interval \([-\pi, \pi]\). By comparing these graphs, you can determine where the graph of \( f(x) \) is above or equal to the graph of \( g(x) \), thus finding the solution to the inequality. Graphical solutions provide an intuitive method of understanding inequalities, simplifying the process of finding intervals where the inequality holds true.

Intersection points

Intersection points are crucial in understanding where the behavior of the inequality switches between true and false. These points occur where the graphs of \( f(x) \) and \( g(x) \) meet. Finding these involves solving the equation:

• \( \cos(2x - 1) + \sin(3x) = \sin\left(\frac{1}{3}x\right) + \cos(x) \)

These points represent values of \( x \) where the trigonometric functions are equal. By identifying these points on the graph, we can split the domain \([-\pi, \pi]\) into distinct intervals. Each interval will need to be tested to see if the inequality \( \cos(2x - 1) + \sin(3x) \geq \sin\left(\frac{1}{3}x\right) + \cos(x) \) holds. Intersection points provide a roadmap for the areas needing further analysis.

Testing intervals

Once the intersection points are determined, the next step is to test the intervals they form. Testing intervals involves picking sample points within each interval and evaluating whether \( f(x) \geq g(x) \). Here's how to approach this:

• Select a test point within each identified interval.
• Substitute this test point into both functions \( f(x) \) and \( g(x) \).
• Check if \( f(x) \) is greater than or equal to \( g(x) \) at that point.

If the inequality holds for the test point, then it generally holds for the entire interval between the intersection points. Testing intervals ensures that we accurately include every segment of \( x \) that satisfies the inequality within \([-\pi, \pi]\). This step confirms which intervals should be combined to form the complete solution set.

Trigonometric functions

Trigonometric functions play a foundational role in solving the given inequality. They involve periodic properties which can create alternating intervals of solutions when plotted over a range like \([-\pi, \pi]\). In the exercise, the functions involved are:

• \( \cos(2x - 1) \), which compresses and shifts the cosine wave.
• \( \sin(3x) \), which increases the frequency of the sine wave.
• \( \sin\left(\frac{1}{3}x\right) \), which decreases the frequency of the sine wave.
• \( \cos(x) \), the standard cosine wave.

Understanding these functions and their transformations is key to interpreting their graphs. Each function has a unique way of oscillating within the interval, affecting where the graphs intersect and creating intervals of solution to the inequality. Recognizing the effects of transformations like frequency and phase shifts helps in understanding why certain values of \( x \) satisfy the inequality while others do not.