Problem 78 - Algebra and Trigonometry

Question

Determine whether the function is even, odd, or neither. $$ f(x)=\cos (\sin x) $$

Step-by-Step Solution

Answer

The function \( f(x) = \cos(\sin x) \) is even.

Step 1: Understand the Definitions

A function \( f(x) \) is **even** if \( f(-x) = f(x) \) for all \( x \). A function is **odd** if \( f(-x) = -f(x) \) for all \( x \). A function is **neither** if it does not satisfy either condition.

Step 2: Substitute -x into the Function

Calculate \( f(-x) \):\[ f(-x) = \cos(\sin(-x)) \]Since \( \sin(-x) = -\sin(x) \), we have:\[ f(-x) = \cos(-\sin(x)) \]

Step 3: Use the Property of Cosine Function

Recall that cosine is an even function, which means \( \cos(-y) = \cos(y) \). Thus:\[ \cos(-\sin(x)) = \cos(\sin(x)) \]So, \( f(-x) = \cos(\sin(x)) = f(x) \).

Step 4: Draw Conclusion

Since \( f(-x) = f(x) \) holds for \( f(x) = \cos(\sin(x)) \), the function is even.

Key Concepts

Symmetry in Mathematics

Symmetry is a fundamental concept in mathematics that describes a situation where one part of a figure or graph is a mirror image of another. This is not just about visual patterns but also about equations and functions.

In the world of functions, symmetry can help determine whether a function is classified as even, odd, or neither. Even symmetry means that a function does not change when its input is replaced by its negative. Mathematically, this is given by the condition:

• Even Function: \( f(-x) = f(x) \)

On the other hand, odd symmetry implies that a function flips its sign when the input changes sign:

• Odd Function: \( f(-x) = -f(x) \)

If a function doesn't meet these criteria, it is considered neither. Identifying symmetry in a function is crucial as it simplifies graphing and analysis. It also leads to understanding the behavior of functions over specific intervals or domains.

In practical terms, symmetry can reveal deep insights into the inherent properties of functions, providing shortcuts and strategies for solving mathematical problems.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is one of the critical trigonometric functions. It is often associated with the measurement of angles and appears in periodic wave patterns such as sound waves.

A remarkable property of the cosine function is its even nature. This means that when you take the cosine of a negative angle, you will get the same result as taking the cosine of the positive angle. Mathematically, this is expressed as:

• Cosine's Evenness: \( \cos(-y) = \cos(y) \)

This property is significant when determining the symmetry of more complex functions that involve the cosine component. For instance, in our original exercise, analyzing the function \( f(x) = \cos(\sin(x)) \) showcases this even symmetry. Because cosine is even, the outcome does not change when the sign of its input changes, as demonstrated by:

• \( \cos(-\sin(x)) = \cos(\sin(x)) \)

This understanding is vital for solving symmetry-related problems as it allows us to quickly determine the nature of the function without extensive computation.

In broader applications, the cosine function's properties, especially its graph’s symmetry, make it an invaluable tool in physics, engineering, and other scientific disciplines.

Sine Function

The sine function, represented as \( \sin(x) \), is another primary trigonometric function that is fundamental in various mathematical and real-world applications. Unlike the even cosine function, the sine function is classified as odd.

To say a function is odd means that if you replace its argument with its negative, the function itself becomes negative. For the sine function, this is represented by:

• Sine's Odd Nature: \( \sin(-y) = -\sin(y) \)

This property of the sine function plays a key role in understanding the composition of functions, like in the given exercise \( f(x) = \cos(\sin(x)) \). Here, when substituting \(-x\) for \(x\), the sine function converts \( \sin(-x) \) to \(-\sin(x)\), illustrating its odd nature.

The odd symmetry of the sine function provides insights when graphing and analyzing periodic phenomena, such as oscillations and waves in physics. Its graph is symmetric about the origin, reflecting its intrinsic symmetrical property over its domain. This understanding is essential in fields like acoustics and electronics, where the sine function models harmonic and sinusoidal movements.