Problem 74 - Algebra and Trigonometry

Question

Use a graphing device to find the solutions of the equation, correct to two decimal places. $$\cos x=\frac{x}{3}$$

Step-by-Step Solution

Answer

Solutions occur at the intersection points of \( y = \cos x \) and \( y = \frac{x}{3} \) on the graph.

Step 1: Understand the equation

We need to find the solutions to the equation \( \cos x = \frac{x}{3} \) using a graphing device. This equation involves both a trigonometric function and a linear function.

Step 2: Graph the functions

Graph the two functions \( y = \cos x \) and \( y = \frac{x}{3} \) on the same coordinate plane using a graphing calculator, software, or plotting tool. Ensure the graph includes both positive and negative values of \( x \) as the cosine function is periodic.

Step 3: Identify the intersection points

Observe the graph for points where the two functions intersect. These points of intersection correspond to the solutions of the equation \( \cos x = \frac{x}{3} \). You may need to zoom in or adjust the window to observe these points precisely.

Step 4: Find the x-coordinates of intersection

Using the graphing tool's intersection feature or by examining the graph, determine the approximate \( x \)-coordinates of the intersection points. Ensure that these values are accurate to two decimal places.

Step 5: Verify the solutions

Check each solution by substituting the \( x \)-values back into the original equation \( \cos x = \frac{x}{3} \) to ensure both sides are approximately equal at these points. This reaffirms that the intersections are correct solutions.

Key Concepts

Intersection Points

Intersection points occur where two functions on a graph meet or cross each other. In mathematical terms, these are the values of the variable, usually denoted as \( x \), where the outputs of the two functions are equal. For example, if we have the trigonometric equation \( \cos x = \frac{x}{3} \), the intersection points are the values of \( x \) where both expressions produce the same result. To find these points using a graphing tool, follow these steps:

• Input both functions into your graphing device:
   - The cosine function \( y = \cos x \)
   - The linear function \( y = \frac{x}{3} \)
• Look for where the graphs of these two functions meet.
  These are your intersection points.

After pinpointing these locations, ensure the coordinates are noted precisely, generally to two decimal places, for accuracy. Remember, these points reveal where the two functions agree entirely.

Graphing Functions

Graphing functions is an essential mathematical skill that visualizes the behavior of equations. This process is crucial when solving equations that involve more than one type of function, such as \( \cos x = \frac{x}{3} \). Here's how to graph these:

• Choose a graphing tool. This could be a graphing calculator, software, or an online plotting tool.
  Each has different features, but they all serve the same fundamental purpose.
• Plot the functions separately.
  For the cosine function, focus on its periodic nature, and for the linear function, pay attention to its non-repetitive progress.
• Adjust the viewing window to capture both positive and negative regions of \( x \).
  This helps in understanding the entire function's behavior.

By graphing these functions on the same plane, you can visually identify where they intersect, offering insight into potential solutions to the equation. This visual approach simplifies understanding, especially when dealing with complex equations.

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It is a periodic function, meaning it repeats its values at regular intervals. Specifically, the cosine function has a period of \( 2\pi \) radians, or 360 degrees. This quality makes it useful in modeling cyclic behaviors, such as waves or circular movements.It's essential to recognize the characteristics of the cosine function when solving equations like \( \cos x = \frac{x}{3} \):

• The function oscillates between -1 and 1.
• Critical points such as maxima, minima, and zeros appear regularly due to periodicity.
• The graph is symmetrical around the y-axis, which is known as even symmetry.

Understanding these properties allows you to anticipate the behavior of the cosine function as it interacts with other equations, like a linear function, in graphing contexts.

Linear Function

A linear function is a function of the form \( y = mx + b \), representing a straight line when plotted on a coordinate plane. In the given equation \( \cos x = \frac{x}{3} \), the linear portion \( y = \frac{x}{3} \) involves a slope of \( \frac{1}{3} \) and no y-intercept. This means the line crosses through the origin and rises steadily but at a gentle angle.Notable aspects of linear functions include:

• Straight-line graphs making them predictable and easy to draw.
• Constant slope, indicating uniform rate of change, unlike the fluctuation seen in trigonometric functions.
• Extended indefinitely in both the positive and negative x-directions, allowing broad intersections with periodic functions like cosine.

In the process of solving equations, understanding the straightforward nature of linear functions can aid in predicting where they might intersect with more complex functions. This intersection is key in solving equations and verifies practical understanding of mathematical concepts.