Forty-five degrees, folded from a square

Fold a square of paper along its diagonal. The diagonal joins two opposite corners. Look at the triangle the fold makes.

One fold, three values

The fold makes the two halves match. So the diagonal splits the square's right angle into two equal parts. Each part is 45°.

The triangle has legs 1 and 1. The legs are the two sides beside the right angle.

Pythagoras says leg² + leg² = hypotenuse². Here 1² + 1² = 2. So the hypotenuse, which is the diagonal, is √2 long.

At the 45° corner, opposite is 1 and adjacent is 1. The hypotenuse is √2.

Rule: sin 45° = 1/√2. cos 45° = 1/√2. tan 45° = 1.

Both legs are equal, so sin 45° and cos 45° match. A bigger square gives the same three values. The side length cancels.

Try the lab to finish this lesson.