Using $\triangle ABC$ below to complete the following problems.

1. Find the exact length of $\overline{A B}$ in terms of $n$.

   Steps (a):

   $AB^2 = (n\sqrt{2})^2 + (n\sqrt{2})^2$

   $AB^2 = 2n^2+2n^2$

2. In terms of $n$, find $\sin A$.

   Hint (b):

   $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$

3. If $n = 12$, what is the value of $\sin A$?

   Answer (c):

   ${\frac{\sqrt{2}}{2}}$

4. What does this tell you about $\sin45^\circ$?

   Answer (d):

   It does not matter what the side length is as long as $A = 45^\circ$.

5. What about $\cos45^\circ$?

   Hint (e):

   $\cos\left(45^{\circ}\right)=\sin\left(45^{\circ}\right)$