The maximum value of the function $y = \sin(x)$ is $1$.

To comprehend why the highest value of $y = \sin(x)$ is $1$, it is essential to examine the characteristics of the sine function. The sine function is periodic, meaning it exhibits a repeating pattern at regular intervals. Specifically, it has a period of $360^\circ$ (or $2\pi$ radians), indicating that the function’s values repeat every $360^\circ$.

The sine function oscillates between $-1$ and $1$. Consequently, for any angle $x$, the value of $\sin(x)$ will always fall within this range. The maximum point on the sine curve is $1$, which occurs at specific angles within each period. For instance, $\sin(90^\circ) = 1$ and $\sin(450^\circ) = 1$. In radian measure, this translates to $\sin\left(\frac{\pi}{2}\right) = 1$ and $\sin\left(\frac{5\pi}{2}\right) = 1$.

To visualize this behavior, consider sketching the graph of $y = \sin(x)$. The graph forms a smooth, wave-like curve, oscillating between $1$ and $-1$. The points at which the curve reaches $1$ correspond to the peaks of the wave. These peaks occur at regular intervals, specifically at:

where $k$ is any integer.

By understanding these properties, it becomes clear that the maximum value of the sine function is always $1$, regardless of the angle $x$. This characteristic is fundamental to the sine function and plays a crucial role in solving various trigonometric problems in GCSE Mathematics.