Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x+\frac{\pi}{4}\right)+\cos \left(x-\frac{\pi}{4}\right)=1$$

Step-by-Step Solution

Answer

Approximate solutions to the equation on the interval \([0, 2\pi)\) are: \(x = 0.78\) and \(x = 5.50\).

Step 1: Set Up the Equation

The given equation is \(\cos \left(x+\frac{\pi}{4}\right)+\cos \left(x-\frac{\pi}{4}\right)=1\). As per the trigonometric identity, it can be rewritten as \(2 \cos (x) \cos \left(\frac{\pi}{4}\right) = 1\) or \(\cos (x) = \frac{1}{2\cos \left(\frac{\pi}{4}\right)}\), which simplifies to \(\cos (x) = \frac{\sqrt{2}}{2}\) . This is the equation we will solve graphically.

Step 2: Plot the Graph

Use a graphing utility to plot the function \(\cos (x)\) and horizontal line \(y=\frac{\sqrt{2}}{2}\) on the interval \([0,2 \pi)\). It particularly concerns the points where the graph of the function crosses the horizontal line, those are the solutions to our equation.

Step 3: Extract the solutions

Inspect the graph for the intersection points of \(\cos(x)\) and \(y=\frac{\sqrt{2}}{2}\), which represent the solutions to our equation within our given interval \([0,2 \pi)\). The graph intersection points approximatly occur at \(x = 0.78\) and \(x = 5.50\), so those are our solutions.

Key Concepts

Understanding Graphing Utility

Graphing utilities, like online graphing calculators or software, are great tools for visualizing mathematical functions. They help you easily observe how a function behaves across a specific interval. When solving trigonometric equations, using a graphing utility can simplify the process of finding where two graphs intersect.
To use a graphing utility:

• Input the equation or function you want to analyze.
• Set the appropriate interval (in this case, \( [0, 2\pi ) \)).
• Plot any horizontal lines you need for comparison.

Intersection points show solutions to the equation, making visualization a powerful tool in math.

The Cosine Function Explained

The cosine function is one of the primary trigonometric functions. It describes the ratio of the adjacent side over the hypotenuse in a right triangle. When graphed, \( \cos(x) \) creates a wave pattern known as a cosine wave, repeating every \( 2 \pi \).
Key aspects of the cosine function include:

• Its maximum value is 1.
• Its minimum value is -1.
• It is periodic with a period of \( 2\pi \).

In this exercise, we compare \( \cos(x) \) with a constant value to find its points of intersection.

Decoding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. They help in simplifying and solving equations.
Important trigonometric identities include:

• Pythagorean identities
• Angle sum and difference identities
• Double angle identities

In our example, the angle difference identities allow us to simplify the original equation:\[\cos(x + \frac{\pi}{4}) + \cos(x - \frac{\pi}{4}) = 2 \cos (x) \cos (\frac{\pi}{4})\]These identities make it easier to rewrite and solve complex trigonometric equations.

Finding Interval Solutions

Interval solutions focus on determining where within a specific range a trigonometric equation has solutions. We explore intervals like \( [0, 2\pi) \), aiming to find "x" values that satisfy our equation.
When working with interval solutions:

• Identify the interval of interest.
• Use graphing utilities to observe intersections within the interval.
• Approximate the 'x' values at these points.

In this problem, by checking intersections within \( [0, 2\pi) \), we find solutions at approximately \( x = 0.78 \) and \( x = 5.50 \). This method helps in solving equations constrained by specific interval limits.