Proofs of the addition formulae for angles of any magnitudes are sometimes rendered invalid by the way in which arguments are founded on a diagram which can represent only one of the many possible cases. When the two angles under consideration are represented in the usual manner as having a common vertex, the only angles in the diagram whose values it is safe to represent by formulae, if complete generality is to be preserved, are angles having this same vertex. The introduction of other angles may be avoided by a restatement of the definitions of sine and cosine.