### Instruction: Compute $\sin 45^\circ$.### Response: I recognize 45 degrees as a special angle, and I remember that there is a formula for the sine of any angle in a 45-45-90 triangle.
In such a triangle, the hypotenuse is $\sqrt{2}$ times the length of a leg, and the sine of an angle is the opposite side over the hypotenuse.
Therefore, $\sin 45^\circ = \frac{\text{leg}}{\sqrt{2} \times \text{leg}} = \frac{1}{\sqrt{2}}$.
I could also rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$, which would give me $\frac{\sqrt{2}}{2}$, but either form is acceptable.