Is cos x an even or odd function?
Understand the Problem
The question is asking whether the cosine function, denoted as cos x, is classified as an even function or an odd function in mathematics. To determine this, we can use the definitions of even and odd functions. An even function satisfies the property f(-x) = f(x), while an odd function satisfies f(-x) = -f(x).
Answer
even function
Answer for screen readers
The cosine function, $\cos(x)$, is an even function.
Steps to Solve
- Recall definitions of even and odd functions
An even function satisfies the property that $f(-x) = f(x)$, while an odd function satisfies the property that $f(-x) = -f(x)$.
- Apply the definition to cos(x)
Find $\cos(-x)$ and compare it to $\cos(x)$. We know from trigonometric identities that
$$\cos(-x) = \cos(x)$$
Since $\cos(-x) = \cos(x)$, the cosine function meets the criteria for an even function.
- Conclude the classification
Since $\cos(x)$ satisfies the property $f(-x) = f(x)$, we conclude that $\cos(x)$ is an even function.
The cosine function, $\cos(x)$, is an even function.
More Information
The cosine function exhibits symmetry about the y-axis. This means that for any angle $x$, $\cos(x)$ yields the same result as $\cos(-x)$.
Tips
A common mistake is to confuse the properties of even and odd functions. Always remember that for an even function, $f(-x) = f(x)$, and for an odd function, $f(-x) = -f(x)$.