# [[Evenness of the Cosine Function]] (Mathematics > Trigonometry) ## Definition A function $f(x)$ is **even** if it satisfies the property: $f(-x) = f(x)$ This means that the graph of an even function is symmetric about the $y$-axis. The cosine function, $\cos x$, is an even function. This means that for any $x$: $\cos(-x) = \cos(x)$ ## Key Concepts - **Even function**: A function where $f(-x) = f(x)$. - **Symmetry**: The graph of an even function is symmetric about the $y$-axis. - **Cosine as even**: For all $x$, $\cos(-x) = \cos(x)$. ## Important Properties 1. **$y$-axis symmetry**: The cosine function is symmetric about the $y$-axis. 2. **Even identity**: $\cos(-x) = \cos(x)$ for all $x$. 3. **Periodicity**: The cosine function is periodic with period $2\pi$. ## Essential Formulas - Even function property: $f(-x) = f(x)$. - $\cos(-x) = \cos(x)$. ## Core Examples 1. For $x = \frac{\pi}{4}$: - $\cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$. 2. For $x = -\frac{\pi}{3}$: - $\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$. ## Related Theorems/Rules - **Even and odd function properties**: An even function satisfies $f(-x) = f(x)$, while an odd function satisfies $f(-x) = -f(x)$. - **Symmetry**: Even functions exhibit symmetry about the $y$-axis. ## Common Pitfalls - Confusing the even property of cosine with odd functions like sine, where $\sin(-x) = -\sin(x)$. - Forgetting that cosine's symmetry is about the $y$-axis, not the origin. ## Related Topics - [[Trigonometric Functions]] - [[Unit Circle Overview]] - [[Comparison of Sine and Cosine Functions]] - [[Odd Functions]] - [[Reflection of a Function Across the X-Axis]] - [[General Transformations of Functions]] - [[Periodic Functions]] - [[Derivatives of Trigonometric Functions]] - [[Pythagorean Identity]] - [[Reflections of Functions]] - [[Oddness of the Sine Function]] - [[Symmetry of Trigonometric Functions]] ## Quick Review Questions 1. How do you prove that $\cos x$ is an even function? 2. What is $\cos(-\frac{\pi}{6})$ based on the evenness of cosine? *** ![[Evenness_of_the_Cosine_Function_visualization]]