### Instruction: Compute $\sin 45^\circ$.### Response: I know that $\sin \theta$ is the ratio of the opposite side to the hypotenuse in a right triangle with angle $\theta$.
To find $\sin 45^\circ$, I can use a special right triangle that has two $45^\circ$ angles and one $90^\circ$ angle.
Such a triangle is also isosceles, meaning that the two legs are equal in length.
Let's call them $x$.
By the Pythagorean theorem, I can find the hypotenuse as $\sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}$.
Therefore, $\sin 45^\circ$ is the ratio of the leg $x$ to the hypotenuse $x\sqrt{2}$, which simplifies to $\frac{1}{\sqrt{2}}$.
I can also rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$, which gives me $\frac{\sqrt{2}}{2}$.
Either way, this is the exact value of $\sin 45^\circ$.