Given a square piece of paper, prove using paper folding alone this trigonometric identity:

arctan(1/3) + arctan(1) = arctan(2)

*** No geometry instrument aids like straightedge, compass etc are allowed.

For reference call the corners A, B, C, D clockwise.
All folds are immediately unfolded forming creases.

1) Fold C to A forming crease BD.
2) Fold B to A forming crease with E on AB and F on BD and G on CD.
3) Fold along ED forming crease ED.

The equation is proved by the angles BDC + BDE = CDE

since BC=CD, angle BDC = arctan(1)
since EG=2*DG, angle, angle CDE = arctan(2)
so we just need to show angle BDE = arctan(1/3)

All the angles are there we just need this last fold for proof, so I'm considering this half a fold.
3.5) Fold B to F forming a crease though H on BD

triangle EHD is  right with legs DH=3*EG so we are done.

Posted by Jer on 2014-06-18 15:02:57