The range of the function $y = \cos(x)$ is given by the interval $[-1, 1]$.

The cosine function, represented as $\cos(x)$, is a fundamental trigonometric function that relates an angle $x$ to the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. When we discuss the range of a function, we refer to all the possible output values that the function can produce. For the function $y = \cos(x)$, the output values are confined to the interval $[-1, 1]$, inclusive. This indicates that regardless of the angle $x$ that you input into the cosine function, the result will always fall within the range of -1 to 1.

To understand why this restriction exists, consider the unit circle, which is a circle with a radius of $1$ centered at the origin of a Cartesian coordinate plane. The cosine of an angle $x$ corresponds to the x-coordinate of the point on the unit circle associated with that angle. Since the radius of the unit circle is $1$, the x-coordinate (and consequently the cosine value) can never be less than $-1$ or greater than $1$. This geometric perspective clarifies why the range of $y = \cos(x)$ is confined to $[-1, 1]$.

Furthermore, the cosine function exhibits periodicity with a period of $2\pi$. This means that the function’s values repeat every $2\pi$ units along the x-axis. Despite this periodic behavior, the maximum value of $\cos(x)$ remains consistently $1$, while the minimum value is consistently $-1$. This reinforces the notion that the range of the function is indeed limited to $[-1, 1]$. Understanding this property is crucial in various fields, such as wave analysis and signal processing, where recognizing the limits of a function’s output is essential for accurate interpretation and application.