The Proofs of the Law of Tangents
798 SCHOOL SCIENCE AND MATHEMATICS THE PROOFS OF THE LAW OF TANGENTS. BY R. M. MATHEWS, " . Riverside, California. In the triangle ABC let a, /?, y be the angles at the respective vertices and a, b, c the sides opposite. The title "law of tangents" is used to denote the trigonometric formula ab tan (aP) _ 3^ a+b ~~ tan 0+/3) and each of those formed by cyclic^ permutation of the letters. The proofs of this theorem in American textbooks stand in decided contrast to those of the law of sines and law of cosines. The proofs of these are based .directly on the triangle and are such as to suggest the one needed extension in definition; when a is an obtuse angle. sin a,== sin(180a), cos a = cos (180a). These two definitions, be it observed, require no notion of co- ordinates or of quadrants. On the other hand, in ten of a group of twelve American texts, the only proof of the law of tangents is by algebraic manipulation from the law of sines and involv- ing the factor formulas for the sum and difference of two sines. The proofs of these two formulas, being based on the addition formulas, are invariably preceded by the definition of the func- tions for the general angle, reduction to the first quadrant, and proof of the regular list of trigonometric identities. Thus the attainment of a formula that applies to triangles only seems to depend on much formal development of general and analytic trigonometry. The law of tangents can be proved directly from a figure, however.
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