Problem 62 - Precalculus with Calculus Previews

Question

By graphing determine whether the given equation has any solutions. $$ \sin x=x $$

Step-by-Step Solution

Answer

The equation \( \sin x = x \) has one solution at \( x = 0 \).

Step 1. Understand the Problem

The equation \( \sin x = x \) asks us to find if there are any points where the sine of \( x \) is equal to \( x \) itself. To find a solution, we need to determine if there are any x-values where the graph of \( \sin x \) intersects the graph of \( y = x \).

Step 2. Sketch the Graph of \( \sin x \)

Draw the graph of \( y = \sin x \), which is a periodic wave that oscillates between -1 and 1 with a period of \( 2\pi \). This wave pattern repeats indefinitely in both the positive and negative x-directions.

Step 3. Sketch the Graph of \( y = x \)

Plot the line \( y = x \). This is a straight line passing through the origin (0,0) at a 45-degree angle to both the x and y axes, representing all points where the x-value equals the y-value.

Step 4. Identify Points of Intersection

Look for points where the graph of \( y = \sin x \) intersects with the graph of \( y = x \). The first clear intersection is near \( x = 0 \) since both \( \sin x \) and \( x \) are equal to 0. Check for other intersections by examining where the sine wave meets or crosses the line \( y = x \).

Step 5. Determine Existence of Solutions

By inspecting the graph, note that \( y = \sin x \) only meets \( y = x \) at the origin (0,0) since beyond this point, \( \sin x \) stays within [-1,1], while \( y = x \) continues to increase or decrease. Therefore, there are no other intersections.

Key Concepts

Graphing Techniques

When solving trigonometric equations like \( \sin x = x \), graphing techniques become essential. By plotting both functions, you can visually determine if and where they intersect.

• Function Visualization: Graphing helps you visualize the behavior of different functions, making it easier to comprehend where or if they converge.
• Scaling and Axes: Make sure to scale your x and y axes appropriately to accurately capture the functions involved. The sine function oscillates between -1 and 1, while the line \( y = x \) will extend far beyond this range.

Cross-checking sketches against calculated points can affirm your findings. Graphs not only simplify problem-solving but also enhance understanding of function behaviors.

Periodic Functions

The sine function, \( y = \sin x \), is a classic example of periodic functions, which are functions that repeat values at regular intervals.

• Understanding Periodicity: A periodic function like \( \sin x \) has a constant period—in this case, \( 2\pi \). This means every \( 2\pi \) units along the x-axis, the function repeats its pattern.
• Oscillation Characteristics: The sine function oscillates between -1 and 1. Understanding the amplitude and period is vital when comparing it to other functions like \( y = x \), which increases indefinitely.

Recognizing these characteristics helps pinpoint potential intersections and solve equations involving periodic functions.

Points of Intersection

Determining the points of intersection between two graphs, such as \( y = \sin x \) and \( y = x \), requires close inspection of how and where these graphs meet.

• Visual Identification: By plotting both graphs, intersections can be identified visually. Look for crossing points where both functions have the same output for a given x-value.
• Verification: After identifying potential intersections, verify by substituting the x-values into both equations to see if they truly equalize. For \( \sin x = x \), only the point (0,0) satisfies this condition in the primary cycle.

Analyzing intersections is a powerful technique to solve equations and understand the relationship between different mathematical functions.