### Instruction: Compute $\sin 45^\circ$.### Response: I know that $\sin 45^\circ$ is the same as $\sin \frac{\pi}{4}$ radians, since $45^\circ$ is one-fourth of a right angle.
I also know that $\sin \frac{\pi}{4}$ is the y-coordinate of a point on the unit circle that makes an angle of $\frac{\pi}{4}$ with the positive x-axis.
I can use the Pythagorean theorem to find the coordinates of that point, since it is on the hypotenuse of a right isosceles triangle with legs of length 1.
The hypotenuse has length $\sqrt{1^2 + 1^2} = \sqrt{2}$, so the coordinates are $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$.
Therefore, $\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$.
Alternatively, I could use the formula $\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}$ with $\alpha = \frac{\pi}{2}$ to get the same result.