Problem 74 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand 'Even' and 'Odd' Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \), while an odd function satisfies \( f(-x) = -f(x) \). A function that does not satisfy either condition is neither even nor odd.

Step 2: Calculate \( f(-x) \)

Given \( f(x) = \sin x + \cos x \). To check whether it is even or odd, calculate \( f(-x) \):\[f(-x) = \sin(-x) + \cos(-x)\] Using trigonometric identities, \( \sin(-x) = -\sin x \) and \( \cos(-x) = \cos x \). So,\[f(-x) = -\sin x + \cos x\]

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

Compare \( f(-x) = -\sin x + \cos x \) with \( f(x) = \sin x + \cos x \) and \(-f(x) = -\sin x - \cos x \):- \( f(-x) eq f(x) \) since \(-\sin x + \cos x eq \sin x + \cos x \)- \( f(-x) eq -f(x) \) since \(-\sin x + \cos x eq -\sin x - \cos x \)

Step 4: Conclusion

Since \( f(-x) \) is not equal to \( f(x) \) or \(-f(x) \), the function \( f(x) = \sin x + \cos x \) is neither even nor odd.

Key Concepts

Trigonometric Identities

Trigonometric identities help us transform and simplify expressions involving trigonometric functions such as sine and cosine. In analyzing the function \( f(x) = \sin x + \cos x \), understanding key identities is crucial.
One essential identity is that for any angle \( x \), \( \sin(-x) = -\sin x \) and \( \cos(-x) = \cos x \). These are known as the odd and even properties of sine and cosine, respectively.

• Odd Identity: Odd functions satisfy \( f(-x) = -f(x) \). Sine is an example of an odd function because \( \sin(-x) = -\sin x \).
• Even Identity: Even functions satisfy \( f(-x) = f(x) \). Cosine is an example of an even function because \( \cos(-x) = \cos x \).

By applying these identities, you can determine how the function \( f(x) = \sin x + \cos x \) behaves under negation of the input, which is a first step in understanding whether it's even, odd, or neither.

Function Symmetry

Function symmetry involves analyzing how a function behaves when its input is negated. This helps determine whether a function is even, odd, or neither.
An even function, like \( \cos(x) \), has symmetry about the y-axis, meaning it looks the same on either side of the y-axis. The condition for even functions is \( f(-x) = f(x) \).

• Symmetry about y-axis: Implies \( f(x) = f(-x) \), so even functions mirror across the y-axis.

On the other hand, an odd function, like \( \sin(x) \), has rotational symmetry about the origin, meaning a 180-degree rotation results in the same graph. An odd function satisfies \( f(-x) = -f(x) \).

• Origin Symmetry: Indicates \( f(-x) = -f(x) \), which is common for functions like \( f(x) = x^3 \).

The function \( f(x) = \sin x + \cos x \) was tested for symmetry by evaluating \( f(-x) = -\sin x + \cos x \). Since \( f(-x) \) neither equals \( f(x) \) nor \(-f(x) \), we conclude there is no symmetry of this kind.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions that describe periodic wave patterns. Each function has its specific traits regarding symmetry.
- **Sine Function:** - Represents a wave that starts from the origin, peaks, descends, and repeats. - It is an odd function, demonstrated by \( \sin(-x) = -\sin x \).
- **Cosine Function:** - Resembles a wave that starts at a maximum, decreases, then increases back to the start value. - It is an even function, validated by \( \cos(-x) = \cos x \).
When combined, the expression \( f(x) = \sin x + \cos x \) exhibits a unique pattern unaffected by the typical symmetries of the individual sine and cosine functions. This function produces a waveform that doesn't showcase specific symmetry around the y-axis or about the origin. Ultimately, this characteristic makes \( f(x) = \sin x + \cos x \) neither strictly an even nor an odd function.