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Chapter 4: Problem 62

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

Short Answer

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

Step by step solution

01

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 \( y = x \).
02

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.
03

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.
04

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 \( y = x \).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攊n 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.

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Most popular questions from this chapter

Find the period and the vertical asymptotes of the given function. Sketch at least one cycle of the graph. $$ y=-1+\sec (x-2 \pi) $$

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ 2 \sin \left(t+\frac{\pi}{2}\right) \cos \left(t-\frac{\pi}{2}\right) $$

Find the indicated value without the use of a calculator. $$ \sec \frac{29 \pi}{4} $$

Find the period, \(x\) -intercepts, and the vertical asymptotes of the given function. Sketch at least one cycle of the graph. $$ y=\cot 2 x $$

Find the period, \(x\) -intercepts, and the vertical asymptotes of the given function. Sketch at least one cycle of the graph. $$ y=-\cot \frac{\pi x}{3} $$
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