Problem 80 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The function is even.

Step 1: Review Symmetry 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: Compute \( f(-x) \)

Given \( f(x) = \cos(\sin x) \), compute \( f(-x) = \cos(\sin(-x)) \). Since \( \sin(-x) = -\sin(x) \), we have \( f(-x) = \cos(-\sin x) \).

Step 3: Simplify \( f(-x) \)

Use the trigonometric identity \( \cos(-\theta) = \cos(\theta) \). Therefore, \( \cos(-\sin x) = \cos(\sin x) \), which means \( f(-x) = \cos(\sin x) = f(x) \).

Step 4: Determine Function Type

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

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving trigonometric expressions. These identities are equations that hold true for all values of the variables involved. They allow mathematicians to establish relationships between different trigonometric functions.

For example, in the original exercise, the identity \( \cos(-\theta) = \cos(\theta) \) plays a crucial role. This identity shows that the cosine function is even, meaning it does not change when the input angle is negated. This property helps in simplifying expressions and finding symmetries in trigonometric functions.

• Certain fundamental identities:
• The Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle-Sum Identities: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• Double Angle Identities: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

These identities simplify the computation of angles and are leveraged in proving the symmetry of trigonometric functions.

Function Symmetry

In mathematics, understanding the symmetry of a function is critical to determining whether it is an even, odd, or neither. Symmetry gives insight into the behavior of functions across the y-axis or the origin.

1. An **even function** satisfies the condition \( f(-x) = f(x) \), indicating symmetry about the y-axis. For the given function \( f(x) = \cos(\sin x) \), calculating \( f(-x) = \cos(\sin(-x)) \), and utilizing the identity \( \sin(-x) = -\sin(x) \), results in \( \cos(-\sin x) = \cos(\sin x) \). Hence, \( f(-x) = f(x) \), showing this function is even.2. An **odd function** satisfies \( f(-x) = -f(x) \), showing symmetry about the origin. If a function does not satisfy either symmetry, it is neither even nor odd.

Most trigonometric functions like sine and tangent are odd, while cosine is even. Recognizing these symmetries aids in simplifying problems that involve even or odd function properties.

Cosine and Sine Functions

Cosine and sine are fundamental trigonometric functions with unique properties and applications.

### Cosine FunctionThe cosine function, denoted as \( \cos(x) \), measures the horizontal coordinate of a point on the unit circle for a given angle \( x \). It is periodic with a period of \( 2\pi \).
Cosine is interesting because it is an **even function**, meaning \( \cos(-x) = \cos(x) \). This property makes it symmetric about the y-axis.

### Sine FunctionThe sine function, denoted as \( \sin(x) \), measures the vertical coordinate of a point on the unit circle for the angle \( x \). Unlike cosine, sine is an **odd function**, with \( \sin(-x) = -\sin(x) \). This property makes sine symmetric with respect to the origin.

Both functions are crucial in calculus, physics, and engineering and are used to model periodic phenomena such as waves and oscillations. Understanding these functions and their identities ensures a robust foundation in trigonometry, aiding in problem-solving and mathematical reasoning.