Problem 85 - Precalculus with Limits

Question

In Exercises \(85-88,\) use a graphing utility to approximate the solutions 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

The solutions are approximately \( x = 0.78\), \( x = 3.92\), within the interval \( \left[0,2\pi\right)\). These are the x-values where the two graphs intersect with the y-value being \(1\).

Step 1: Input the Functions

First, we need to input the function \( y = \cos\left(x + \dfrac{\pi}{4}\right) + \cos\left(x - \dfrac{\pi}{4} \right) \) into the graphing utility. We also need to input the function\( y = 1 \) as a horizontal line.

Step 2: Set the Range

Now we will set the range from 0 to \(2\pi\) because that's the interval we are interested in. We also set a suitable range for the y-coordinate, like -3 to 3, to clearly visualize the functions.

Step 3: Find the Intersection Points

Then, we should look for points where the two graphs intersect. This corresponds to our x-values for which the original equation holds true.

Step 4: Approximate the Solutions

Lastly, we approximate the solutions by reading off the x-coordinates of the intersection points. The intersection points are the x-values that satisfy our equation.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all the allowed values of the variables they contain. These identities are crucial for simplifying trigonometric expressions and solving trigonometric equations. For instance, the sum-to-product identities can transform the sum of two trigonometric functions into a product, which can often be easier to analyze and solve.

For example, the given problem \( y = \cos(x + \frac{\pi}{4}) + \cos(x - \frac{\pi}{4}) \) can be simplified using the sum-to-product identities. This identity states that \( \cos A + \cos B = 2 \cos(\frac{A + B}{2}) \cdot \cos(\frac{A - B}{2}) \) which could potentially simplify the problem and make it more transparent to solve, however, this is less straightforward in the case of sum of cosines with different arguments. Understanding these identities is crucial as they underpin the expressions and equations we encounter in trigonometry.

Graphing Utility

A graphing utility, like a graphing calculator or software, is an invaluable tool for visualizing and understanding the behavior of trigonometric functions. When equations become complicated, a graphing utility provides a visual representation that can help users intuitively grasp when and where functions intersect, for example.

To use a graphing utility effectively, you need to correctly input the function, set a suitable viewing window (in this case, from \(0\) to \(2\pi\)) and identify important features of the graph such as intercepts, maxima, and minima. Graphing utilities also allow for zooming and tracing, which help in pinpointing more accurate intersection points, crucial for the next step, which involves approximating solutions.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy the equation. These angles, or solutions, are usually within a specific interval, such as \( [0,2\pi) \) in our exercise. The fundamental approach to solving trigonometric equations often begins with using trigonometric identities to simplify the equations into a more solvable form.

However, not all trigonometric equations are solvable by simple algebraic means, especially when the solutions are not one of the standard trigonometric angles. In such cases, numerical methods or graphical techniques using a graphing utility are employed to approximate the solutions. Understanding how to rewrite and manipulate these equations is key to identifying possible solutions.

Approximate Solutions

When exact analytical solutions to trigonometric equations are not feasible, we resort to approximate solutions. This is often done with the help of a graphing utility where we visually identify where two functions intersect. Following this, we estimate the x-coordinates of these intersection points to approximate the solutions.

In practice, most graphing utilities will allow us to 'trace' along the curve and read approximate values to a certain number of decimal places. While these solutions are not exact, they are often sufficient for practical purposes, and understanding how to interrogate the graph to extract these approximations is a valuable skill in solving trigonometric problems.