Problem 80 - 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: Define function types

First, let's remember the definitions. A function is even if \( f(-x) = f(x) \) for all \( x \) in its domain and odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition is met, the function is neither even nor odd.

Step 2: Apply even function test

Now, test the function \( f(x) = \cos(\sin x) \). Calculate \( f(-x) \): \( f(-x) = \cos(\sin(-x)) \). Since \( \sin(-x) = -\sin(x) \), it follows that \( f(-x) = \cos(-\sin x) \).

Key Concepts

Function Parity

Function parity is a fundamental property that helps classify functions based on their symmetry. Understanding this can simplify many mathematical processes. There are two main types of function parity: even and odd functions.

• **Even Functions**: A function is considered even if it satisfies the condition that for every input \( x \) in its domain, \( f(-x) = f(x) \). This implies that the graph of an even function is symmetric with respect to the y-axis.
• **Odd Functions**: Conversely, a function is classified as odd if \( f(-x) = -f(x) \) holds true for all \( x \) in its domain. The graph of an odd function is symmetric with respect to the origin.
• **Neither**: If neither condition is met, the function is neither even nor odd, which means it lacks this particular symmetry.

These definitions are integral in various fields, particularly in Fourier analysis, where even and odd function properties simplify calculations.

Trigonometric Functions

Trigonometric functions are functions related to angles and are typically periodic. The basic trigonometric functions include sine, cosine, and tangent among others. Recognizing the parity of trigonometric functions can aid in understanding more complex expressions.

• **Sine Function \( \sin(x) \)**: This function is odd, which means \( \sin(-x) = -\sin(x) \). Its graph exhibits origin symmetry, making it useful for modeling periodic phenomena.
• **Cosine Function \( \cos(x) \)**: Cosine is an even function since \( \cos(-x) = \cos(x) \). It shows symmetry along the y-axis and is also pivotal in wave-based applications.

Trigonometric functions play a crucial role in geometry, physics, and engineering.

Cosine Function

The cosine function, denoted by \( \cos(x) \), is one of the most fundamental trigonometric functions. It is periodic with a period of \( 2\pi \), meaning its pattern repeats every \( 2\pi \) units.

• **Even Nature**: As an even function, \( \cos(x) \) satisfies \( \cos(-x) = \cos(x) \). Its y-axis symmetry simplifies many calculations in trigonometry and calculus.

When considering the function \( \cos(\sin x) \), we involve a composition of functions. Since \( \sin(x) \) is odd, \( \sin(-x) = -\sin(x) \), causing a shift in symmetry when input into \( \cos(x) \). Consequently, this impacts the parity of \( \cos(\sin x) \), potentially making it neither purely even nor odd. Understanding these relationships aids in deeper analysis and application of trigonometric functions.