Geometry of Addition and Subtraction Formulas

These figures show proofs of trigonometric addition and subtraction formulas for cosine and sine by Leonard M. Smiley, University of Alaska Anchorage. The subtraction formulas appeared in the December, 1999 issue of Mathematics Magazine.

The proofs are valid for "small" angles only, but are completely synthetic and use a common, minimal diagram from Euclidean geometry. The claim of minimality is supported by the absence of any unused points and the low number (5 each) of points and segments. The proofs share nothing with the standard analytic proofs using Cartesian coordinates, and thus are fully complementary to these more general treatments.