### Instruction: Compute $\sin 45^\circ$.### Response: To find the sine of an angle, I need to relate it to a right triangle with that angle.
One way to do that is to use a 45-45-90 triangle, which is a special type of isosceles right triangle.
In such a triangle, the two legs have the same length, and the hypotenuse has a length equal to the leg times $\sqrt{2}$.
I can label the sides of such a triangle as $x$, $x$, and $x\sqrt{2}$, where $x$ is any positive number.
Then, by definition, $\sin 45^\circ$ is the ratio of the opposite side to the hypotenuse, or $\frac{x}{x\sqrt{2}}$.
To simplify this fraction, I can divide both the numerator and the denominator by $x$, and get $\frac{1}{\sqrt{2}}$.
This is a valid answer, but I can also rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{2}$.