### Instruction: Compute $\cos 45^\circ$.### Response: I remember that there is a special right triangle with a 45-45-90 degree angle measure, and that the lengths of the sides are in the ratio 1:1:$\sqrt{2}$.
That means if I draw such a triangle and label the hypotenuse as $\sqrt{2}$ and the other two sides as 1, then I can use the definition of cosine to find $\cos 45^\circ$.
Cosine is the ratio of the adjacent side to the hypotenuse, so in this triangle, $\cos 45^\circ$ is 1 over $\sqrt{2}$.
But I know that it is not good form to leave a radical in the denominator, so I can rationalize the fraction by multiplying the top and bottom by $\sqrt{2}$.
That gives me $\sqrt{2}$ over 2 as the simplified expression for $\cos 45^\circ$.