Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin \frac{x}{2}+\cos 3 x=0$$

Step-by-Step Solution

Answer

The solutions are approximately \(x \approx 0.524\) and \(x \approx 5.236\).

Step 1: Understand the Problem

You need to solve \( \sin \left( \frac{x}{2} \right) + \cos 3x = 0 \) over the interval \([0, 2\pi)\). This means you want to find all \(x\) values in this interval that satisfy the equation.

Step 2: Recognize the Components

First, identify the components involved: \( \sin \left( \frac{x}{2} \right)\) and \(\cos 3x\). We will graph these two functions separately to find their intersection points.

Step 3: Sketch the Graph

Graph \(y_1 = \sin \left( \frac{x}{2} \right)\) and \(y_2 = -\cos 3x\) separately. These graphs will capture the behavior of these trigonometric functions within the given interval.

Step 4: Find the Intersection Points

Look for points where \(y_1\) and \(y_2\) intersect, as these points satisfy \( \sin \left( \frac{x}{2} \right) + \cos 3x = 0 \). Use a graphing calculator or software to find these intersections precisely within the interval \([0, 2\pi)\).

Step 5: Round to the Nearest Thousandth

After finding the intersection points, round the x-values of these points to three decimal places, since the problem asks for values rounded to the nearest thousandth.

Key Concepts

Trigonometric Equations

Trigonometric equations are those that involve trigonometric functions like sine, cosine, and tangent. In this exercise, the equation \( \sin \left( \frac{x}{2} \right) + \cos 3x = 0 \) involves both sine and cosine functions. These equations are solved over a specific interval, here being \([0, 2\pi)\), meaning we look for values of \(x\) within one complete cycle of the trigonometric circle.

To tackle such equations effectively, it's crucial to understand how these trigonometric functions behave within the given interval. Sine and cosine functions have certain characteristics, such as periodicity and specific points where they reach maximum, minimum, and zero values. Knowing these attributes helps predict where these functions might intersect. For instance, the sine function \(\sin \left( \frac{x}{2} \right)\) has a half-cycle over the interval \([0, 2\pi)\), while \(\cos 3x\) completes three full cycles. This variation in speed and frequency is essential to uncovering where the equation equals zero, which is at their intersection points.

Graphical Visualization

Graphical visualization is a powerful method to solve trigonometric equations, providing a clear picture of where functions meet or when they sum to zero. By plotting each function separately, \(y_1 = \sin \left( \frac{x}{2} \right)\) and \(y_2 = -\cos 3x\), we transform the equation into a visual problem.

Two major steps are involved in graphical visualization:

• Sketching the Graphs: Begin by sketching the graphs of each function. This step can often be aided by graphing calculators or software, providing accurate representations.
• Analyzing the Behavior: By viewing these graphs together within the interval, you observe how the different periodicities and amplitudes interact, pointing to where they intersect.

These intersections are critical as they show the values of \(x\) that satisfy the equation \(\sin \left( \frac{x}{2} \right) + \cos 3x = 0\). The graphical method not only aids in finding solutions but also enhances understanding of how these trigonometric functions behave in relation to each other.

Intersection Points

Intersection points in the context of solving trigonometric equations are the key to finding the values of \(x\) that satisfy the given equation. These are points on the graphs where \(y_1 = \sin \left( \frac{x}{2} \right)\) and \(y_2 = -\cos 3x\) meet, equating their outputs and satisfying the original equation.

Finding these intersection points involves:

• Identifying Common Points: Once the graphs are plotted, locate where the lines or curves intersect. These intersections represent the solutions to the equation.
• Precision Tools: Using tools like graphing calculators helps in determining the exact values of \(x\) where intersections occur. This is particularly useful when the exact intersection points are not easily discernible by eye.
• Rounding: After determining these points, the exercise often requires rounding these solutions to a specific decimal place, here the nearest thousandth.

This graphical method emphasizes comprehension of trigonometric behavior and provides not just a solution, but a deeper understanding of the interaction between different trigonometric functions.