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

To understand why $\sin 90^\circ$ equals $1$, we must explore the fundamentals of trigonometry and the unit circle. The sine function is defined in relation to the $y$-coordinate of a point on the unit circle, which is a circle with a radius of $1$ centered at the origin of a coordinate plane. When measuring angles in standard position—beginning from the positive $x$-axis and moving counterclockwise—an angle of $90^\circ$ corresponds to a point located directly above the origin on the unit circle.

At $90^\circ$, the coordinates of this point are $(0, 1)$. The sine of an angle, as defined in the context of the unit circle, is simply the $y$-coordinate of the corresponding point. Therefore, at $90^\circ$, the $y$-coordinate is $1$, which means $\sin 90^\circ = 1$.

Another perspective for understanding this is through the lens of a right-angled triangle. If we visualize a right-angled triangle with one of its angles measuring $90^\circ$, the side opposite this angle serves as the hypotenuse. The sine of an angle in a right-angled triangle is calculated as the ratio of the length of the opposite side to the length of the hypotenuse. For an angle of $90^\circ$, the opposite side is the hypotenuse itself, resulting in a ratio of $1$ (since any number divided by itself equals $1$).

Thus, both the unit circle and right-angled triangle perspectives affirm that the exact value of $\sin 90^\circ$ is indeed $1$.