Problem 64 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. \(\sin |x|=\sin x\) for \(-2 \pi < x < 2 \pi\)

Step-by-Step Solution

Answer

False; \( \sin |x| \neq \sin x \) for negative \( x \).

Step 1: Understanding the Problem

We need to determine if the equation \( \sin |x| = \sin x \) holds true for every value of \( x \) in the interval \( -2\pi < x < 2\pi \). This means checking whether the sine of the absolute value of \( x \) is equal to the sine of \( x \) for each value in the given range.

Step 2: Exploring Properties of the Sine Function

The function \( \sin(x) \) is an odd function, meaning \( \sin(-x) = -\sin(x) \). Also, \( \sin(x) \) is periodic with a period of \( 2\pi \), so \( \sin(x) = \sin(x + 2\pi k) \) where \( k \) is an integer. The absolute value transformation \( |x| \) affects the sign of \( x \), which must be considered.

Step 3: Analyzing Positive and Negative x-values

Let's analyze the behavior of \( \sin |x| \) and \( \sin x \):- **For positive values of \( x \) (\( x \geq 0 \)):** - \( |x| = x \), thus \( \sin |x| = \sin x \).- **For negative values of \( x \) (\( x < 0 \)):** - \( |x| = -x \), thus \( \sin |x| = \sin(-x) = -\sin(x) \).From this, we see that for negative \( x \), \( \sin |x| \) is not equal to \( \sin x \).

Step 4: Validating the Result with Examples

Let's check specific examples:- **Example 1:** Let \( x = -\pi/2 \) - \( \sin |x| = \sin(\pi/2) = 1 \) - \( \sin x = \sin(-\pi/2) = -1 \) - Clearly, \( \sin |x| eq \sin x \).- **Example 2:** Let \( x = \pi/4 \) - \( \sin |x| = \sin(\pi/4) = \sin(\pi/4) \) - \( \sin x = \sin(\pi/4) \) - Here, \( \sin |x| = \sin x \).These examples confirm the earlier analysis.

Step 5: Conclusion

From the analysis, we conclude that the statement \( \sin |x| = \sin x \) is false for the interval \( -2\pi < x < 2\pi \). The equation holds only for non-negative values of \( x \), while it fails for negative values, where \( \sin |x| = -\sin x \).

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is represented as \( \sin(x) \) and relates to the ratio of the opposite side to the hypotenuse in a right triangle. The sine function can take any real number as an input, and its range is always between -1 and 1. This means that no matter what value \( x \) has, \( \sin(x) \) will always output a value between -1 and 1.

Key facts about the sine function include:

• It is continuous and smooth, meaning there are no jumps or breaks in its graph.
• The graph of the sine function has a wave-like shape, called a sine wave.

The sine function is more than just a mathematical curiosity; it has many applications. It is often used in physics, engineering, and signal processing to describe oscillations, waves, and cycles.

Absolute Value

The absolute value of a number is simply that number without regard to its sign. In mathematical terms, it measures the distance of the number from zero on the number line. For example, the absolute value of both -5 and 5 is 5, because they are both five units away from zero.

Represented as \( |x| \), the absolute value function has the following properties:

• It is always non-negative. \( |x| \geq 0 \) for any real number \( x \).
• It affects the argument of functions like sine by negating the negative input values, beneath the evaluation.

In the context of our exercise, the use of the absolute value affects the input to the sine function. This becomes important, especially at negative values, fundamentally changing how the function behaves across different parts of its domain.

Odd Functions

An odd function is a type of mathematical function that exhibits specific symmetry, such that \( f(-x) = -f(x) \) for all \( x \) in its domain. In simple terms, this means if you rotate the graph of the function 180 degrees around the origin, it looks the same.

Important characteristics of odd functions include:

• They are symmetric about the origin. This means the graph to the left of the y-axis is a mirror image of the graph to the right.
• If a polynomial is odd, all its exponents are odd.

The sine function, \( \sin(x) \), is an example of an odd function. Because \( \sin(-x) = -\sin(x) \), it has symmetry around the origin. This property becomes crucial when analyzing functions like \( \sin|x| \) because where \( x < 0 \), \( \sin \) evaluates distinctly different than at \( |x| \).

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. This means that for some positive constant \( T \), the function satisfies \( f(x + T) = f(x) \) for all \( x \).

Characteristics of periodic functions include:

• The repeat interval \( T \) is called the period of the function.
• They are often used to model cyclical phenomena such as seasonal changes, sound waves, or oscillations.

The sine function is periodic with a period of \( 2\pi \). This means that every \( 2\pi \) units, its values repeat. In the context of the exercise, even when absolute value modifies \( x \), this periodicity assists in understanding how values behave universally. The graph of \( \sin(x) \) simply cycles back to the start of a new wave every \( 2\pi \). This property is fundamental in analyzing function equivalences like \( \sin |x| \) versus \( \sin x \).