The exact value of $\sin 90^\circ$ is $1$.

To comprehend why $\sin 90^\circ$ equals $1$, we must explore the fundamentals of trigonometry. The sine function, denoted as $\sin(\theta)$, is a key trigonometric function that connects the angle $\theta$ in a right-angled triangle to the ratio of the length of the opposite side to that of the hypotenuse. Specifically, for any angle $\theta$, the relationship is given by:

When we examine the unit circle—defined as a circle with a radius of $1$ centered at the origin of a coordinate plane—the sine of an angle $\theta$ corresponds to the $y$-coordinate of the point where the terminal side of the angle intersects the circle. For an angle of $90^\circ$, the terminal side forms a vertical line that intersects the unit circle at the point $(0, 1)$. This means the $y$-coordinate is $1$, leading to the conclusion that $\sin 90^\circ = 1$.

Another way to visualize this is by considering a right-angled triangle where one of the angles measures $90^\circ$. In such a triangle, the side opposite the $90^\circ$ angle is the hypotenuse itself. Since the hypotenuse is always the longest side and is equal to itself, the ratio of the opposite side to the hypotenuse (i.e., opposite/hypotenuse) is $1$. Thus, we again find that $\sin 90^\circ = 1$.

By understanding these concepts, we can clearly see why the sine of $90^\circ$ is exactly $1$, reinforcing the important relationships between angles and side lengths in trigonometry.