Problem 67 - Calculus

Question

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

Step-by-Step Solution

Answer

The statement is true for \(-2 \pi < x < 2 \pi\) due to the even nature of the cosine function.

Step 1: Understanding Trigonometric Properties

The cosine function, \( \cos x \), is an even function. This means that \( \cos(-x) = \cos(x) \). This property will help us in comparing \( \cos |x| \) and \( \cos x \).

Step 2: Analyzing \( |x| \)

The notation \( |x| \) represents the absolute value of \( x \), meaning it is always non-negative. Thus, if \( x \) was negative, \(|x|\) would equal \(-x\), whereas if \( x \) was positive, \(|x| \) remains \( x \).

Step 3: Comparing \( \cos |x| \) with \( \cos x \)

For negative values of \( x \), say \( x = -a \) where \( a > 0 \), \( |x| = a \), so \( \cos |x| = \cos(a) \). With the cosine’s even property, this equals \( \cos(-a) = \cos(a) \). For positive \( x \), \( \cos |x| = \cos(x) \) directly. Thus, for all \( x \) in the interval \(-2\pi < x < 2\pi \), the equality holds.

Key Concepts

Even Function

An even function is a type of function that has a specific symmetry property. This symmetry means that the value of the function at a positive point is equal to the value of the function at the corresponding negative point. For example, if you take a function \( f(x) \) that is even, you will find that \( f(-x) = f(x) \) for all \( x \) in its domain.

The graph of an even function is symmetric with respect to the y-axis. This property is important in trigonometry because many trigonometric functions, such as the cosine function, are even functions. Recognizing an even function helps us simplify equations and understand relationships within the function. Additionally, knowing a function is even can help confirm correct problem-solving when working with trigonometric identities and equations.

• An important aspect of even functions is that they make calculations easier when dealing with both negative and positive inputs.
• Examples include the cosine function and the absolute value function when considered over specific symmetrical properties.

Absolute Value

The absolute value of a number is a fundamental concept in mathematics representing the non-negative value of a number without regard to its sign. In other words, it tells you how "far" a number is from zero on the number line, regardless of direction.

The notation \( |x| \) is used to denote the absolute value of \( x \). The definition is:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

Understanding absolute value is crucial when solving equations and expressions involving \( |x| \), for example, when comparing \( \cos |x| \) and \( \cos x \). The absolute value preserves the magnitude but discards any negative sign, setting up conditions for using other properties, like the evenness of the cosine function, effectively.

• Recognizing the role of absolute value helps simplify expressions by focusing only on the size.
• It also allows us to consider only the range or magnitude, not the direction or sign.

Cosine Function

The cosine function, denoted as \( \cos x \), is a fundamental trigonometric function that plays a significant role in mathematics and physics. It describes the relationship between an angle and the lengths of the sides of a right triangle. Specifically, it gives the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

The cosine function is periodic with a period of \( 2\pi \), which means it repeats its pattern every \( 2\pi \) radians. Its graph is a wave that oscillates between -1 and 1, making it useful for modeling wave-like phenomena.

• A key feature of the cosine function is that it is an even function, meaning \( \cos(-x) = \cos(x) \), as discussed above.
• This property is particularly useful in simplifying expressions, such as proving the equivalence of \( \cos |x| \) to \( \cos x \) over certain intervals.

Understanding the even property and periodic nature of cosine enhances our ability to model and solve real-world problems.