1. The problem is to find the exact value of $\sin 60^\circ$. 2. Recall that $60^\circ$ is an angle in an equilateral triangle where all sides are equal and all angles are $60^\circ$. 3. To find $\sin 60^\circ$, we can use the properties of a 30-60-90 right triangle, which is formed by splitting an equilateral triangle in half. 4. In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$, where the side opposite $30^\circ$ is 1, opposite $60^\circ$ is $\sqrt{3}$, and the hypotenuse is 2. 5. By definition, $\sin 60^\circ = \frac{\text{opposite side}}{\text{hypotenuse}}$. 6. Using the side lengths, $\sin 60^\circ = \frac{\sqrt{3}}{2}$. 7. Therefore, the exact value of $\sin 60^\circ$ is $$\sin 60^\circ = \frac{\sqrt{3}}{2}.$$