The exact value of $\sin 45^\circ$ is $\frac{\sqrt{2}}{2}$.

To understand why $\sin 45^\circ$ equals $\frac{\sqrt{2}}{2}$, we can start by examining a right-angled isosceles triangle. In this type of triangle, the two non-right angles are both $45^\circ$. If we assume that the lengths of the two equal sides are each $1$ unit, we can apply the Pythagorean theorem to determine the length of the hypotenuse.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). For our triangle, we have:

Next, the sine of an angle in a right-angled triangle is defined as the length of the opposite side divided by the length of the hypotenuse. For $\sin 45^\circ$, the length of the opposite side is $1$ (one of the equal sides), and the hypotenuse is $\sqrt{2}$. Therefore, we can express this as:

To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{2}$:

Thus, we conclude that the exact value of $\sin 45^\circ$ is $\frac{\sqrt{2}}{2}$.