Problem 246 - Precalculus

Question

For the following exercises, use identities to evaluate the expression. Determine whether the function \(f(x)=\sin x-2 \cos ^{2} x\) is even, odd, or neither.

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Recall Definitions of Even and Odd Functions

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 \( x \). If neither condition is satisfied, the function is neither even nor odd.

Step 2: Substitute \(-x\) into the Function

Substitute \(-x\) into the function: \[ f(-x) = \sin(-x) - 2 \cos^2(-x) \]Using the identities \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \), evaluate the expression:\[ f(-x) = -\sin(x) - 2\cos^2(x). \]

Step 3: Compare \(f(-x)\) with \(f(x)\)

We have given \( f(x) = \sin x - 2 \cos^2 x \) and calculated \( f(-x) = -\sin(x) - 2\cos^2(x) \). Now compare them:- They are neither identical \( (f(-x) eq f(x)) \), which means the function is not even.- They are not negatives of each other \( (f(-x) eq -f(x)) \), which means the function is not odd.

Step 4: Conclusion

Since \( f(x) \) is neither equal to \( f(-x) \) nor to \(-f(x)\), the function \( f(x) = \sin x - 2\cos^2 x \) is neither even nor odd.

Key Concepts

Trigonometric Identities

Trigonometric identities are vital tools that allow us to simplify and evaluate trigonometric expressions. They are equations that hold true for any angle substitutions within their domains. Some fundamental trigonometric identities include the Pythagorean identities, the angle sum and difference identities, and the identities for negative angles.

When dealing with the sine and cosine functions, one of the essential identities to remember is the negative angle identities. These are:

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

By using these identities, we're able to evaluate functions more easily by rewriting them in simpler forms. This ability is particularly helpful in determining the even or odd nature of functions, as trigonometric identities can reveal hidden symmetries within the functions.

Function Evaluation

Function evaluation involves substituting a specific value for the variable in a function and simplifying the resulting expression. In the context of determining if a function is even or odd, we substitute \(-x\) into the function to see how it compares to the original function.

For a function like \( f(x) = \sin x - 2 \cos^2 x \), evaluating \( f(-x) \) requires the application of trigonometric identities:

• Substitute \(-x\) into all occurrences of \(x\)
• Simplify using identities such as \( \sin(-x) = -\sin(x) \)
• Compare the new expression to the original \(f(x)\) and \(-f(x)\) to classify the function

This process highlights how identities not only simplify expressions but also help reveal the underlying nature of trigonometric functions in terms of evenness and oddness.

Cosine and Sine Functions

The sine and cosine functions are fundamental trigonometric functions used to model periodic phenomena. Understanding their characteristics is essential when working with trigonometric expressions.

• The sine function, \( \sin(x) \), is odd, which means \( \sin(-x) = -\sin(x) \).
• The cosine function, \( \cos(x) \), is even, indicating that \( \cos(-x) = \cos(x) \).

These properties are directly tied to the symmetry of their respective graphs:

• Sine graph exhibits symmetry about the origin, which defines its odd nature.
• Cosine graph displays symmetry about the y-axis, characteristic of an even function.

While these characteristics help in classification, any combination of sine and cosine, such as in the function \( f(x) = \sin x - 2 \cos^2 x \), can lead to a function that is neither even nor odd, as verified through function evaluation.