1. **State the problem:** Find the expression and properties of the function $\cos(x)$.\n\n2. **Formula and explanation:** The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle or as the x-coordinate of a point on the unit circle at an angle $x$. It is periodic with period $2\pi$, meaning $\cos(x) = \cos(x + 2\pi)$.\n\n3. **Key properties:**\n- Range: $[-1,1]$\n- Period: $2\pi$\n- Even function: $\cos(-x) = \cos(x)$\n- Important values: $\cos(0) = 1$, $\cos(\pi/2) = 0$, $\cos(\pi) = -1$\n\n4. **Intermediate work:** Evaluating at some points:\n$\cos(0) = 1$\n$\cos(\pi/3) = \frac{1}{2}$\n$\cos(\pi/2) = 0$\n$\cos(\pi) = -1$\n\n5. **Explanation:** The cosine function oscillates smoothly between 1 and -1 as $x$ increases, repeating every $2\pi$. It is useful in modeling waves, oscillations, and circular motion.