az aaea = 1 + a 1-2 + 1.23 + . . ad inf.
for every finite value of a real or imaginary. Writing i a for a, where (after Euler) i =-_ one obtains 1 eia = ( as az 1.2 + . .) az +f ( .2.3 1.2.3.4.5 ')' or :--- cos a + i sin a.
Replacing i by = cos a i sin a.
The product of the last two equations yields 1 = cosia sinta, another result known to be true. Once more, vitt = ei(a+ ft) = cos (a + + i sin (a + /3) ; also, elaetP = (cos a + i sin a) (cos p + i sin 8) = cos a cos P i sin a sin /3 + i(sin a cos /3 + cos a sin P); whence cos (a + P) = cos a cos /3 sin a sin P, and sin (a + /3)= sin a cos p + cos a sin p, two equations of the known addition theorem for sine and cosine. It is indeed a fact that the whole body of trigonometric relations de duced or deducible from the original defini tions, Fig. 1, of the functions, are obtainable analytically from (s) and (c) regarded as definitions, and, like the latter, would then be free from geometric reference. Each of the other functions, direct or inverse, is repre sentable in the form of a series analogous to (s) and (c). Such series may be found in books of trigonometry and of the calculus.
DeMoivre s Theorem.From eia = cos a+s sin a it follows that (cos a+i sin Os; but cos na +i sin na, hence (cos sin of)a cos na +1 sin na, a famous theorem due to De Moivre and known as De Moivre's Theorem. Suppose n = 3, then
cos 3a + s sin 3a = (cos a + i sin a)' = cos' a 3 cos a a + i(3 cos' a sin a sins a) ; whence cos 3a = cos' a 3 cos a sin' a, sin 3a = 3 cog a sin a sin' a. Similarly sines and cosines of any multiple of a may be expressed in terms of the like func tions of the single angle.
Ruler's Formulae. From the relations eia = cos a + i sin a, = cos a sin a, may be found, by addition and subtraction, the Eule rian equations 1 1 cos a = 2 (eia + e--(4), sin a = (eta 2i which are equivalent to (c) and (s) and serve to define sin and cos in terms of imaginary powers of the Napierian base.
Some Curious Relations. Letting a eta = cos a + i sin a, we get eiri= i : squaring,. we obtain ea = 1; also erq and ea'" I = 0. The last is especially noteworthy as in volving the most notable set of five numbers in mathematics: 0, 1, e, In. Further develop ments would lead into the doctrine of Circle Partition (Kreistheilung), which belongs to the Theory of Functions of the Complex Variable, to which the reader is referred. See COMPLEX VARIABLE, THEORY OF FUNCTIONS OF A.