Trigonometric Protractor

Thank you to Patrick of Woodbridge School and Matthew whose contributions form the basis of what is written below.

Even when you pull out the protractor the radius stays as 1. The other two sides keep in proportion.

The numbers appear to be the lengths of the sides of any new triangle drawn, assuming the hypotenuse is 1. They do not change when the triangle is scaled because of Pythagoras's theorem states:
The length of hypotenuse squared = length of one other side squared + length of third side squared. This if the triangle is right-angled.

I made a triangle with sides of 1, 0.8 and 0.6 and an angle of 37 degrees. These numbers worked because the two shorter sides were that proportion of the hypotenuse and so this did not depend on the size of the triangle. Also 0.8 squared plus 0.6 squared makes 1 squared (by Pythagoras theorem). By changing the size of the triangle the hypotenuse always stays as 1 so the sides will stay the same because that is the proportion of the hypotenuse they are. The side nearest 37 degrees (the adjacent) is always 0.8 of the hypotenuse and the opposite side is 0.6.

When I made another triangle the sides were 1, 0.92 and 0.4 and the angle was 24 degrees. The same things were true: the hypotenuse was 1 unit the adjacent side is 0.92 and the opposite side is 0.4 so the adjacent side is nearly as long as the hypotenuse and the opposite side less than half. Also 0.92 squared plus 0.4 squared is 1 squared roughly.

For any triangle you can use this protractor you just need to multiply by the factor that is involved. if the hypotenuse is 5 then the other two sides will be 5 times longer than the protractor says. This will work for any length. These are like the sine and cosine of the angle.

The signs change because of the way you measure the angles as you go around. This is like the problem about the dot.