Cor. 1. If a be an angle so small that a2 and higher powers of a may be neglected when compared with unity, (3) becomes cos a = 1, and (4), sin α = α. If a2, a3 be retained, but higher powers of a be neglected, (3) and (4) give Cor. 2. By dividing (3) by (4), we obtain 2 α5 17a7 + etc. (5) tan α = a + + + - 157. Convergence of the Series. The series (3) and (4) of Art. 156 may be proved to be convergent, as follows: The numerical value of the ratio of the successive pairs of consecutive terms in the series for sin a are Hence the ratio of the (n+1)th term to the nth term is ; and whatever be the value of a, we can take a2 2n (2n+1) n so large that for such value of n and all greater values, this fraction can be made less than any assignable quantity; hence the series is convergent. Similarly, it may be shown that the series for cos a is always convergent. 158. Expansion of cos" in Terms of Cosines of Multiples of 0, when n is a Positive Integer. Also x"= (cos +i sin 0)”=cos n0+i sin në (Art. 153) (2) NOTE. In the expansion of (x+x-1)" there are n+1 terms; thus when n is even there is a middle term, the +1)th, which is independent of 6, and which is n 2 Hence when n is even the last term in the expansion of 2′′-1 cos" is When n is odd the last term in the expansion of 2"-1 cos" "is 159. Expansion of sin" in Terms of Cosines of Multiples of 0, when n is an Even Positive Integer. 160. Expansion of sin" in Terms of Sines of Multiples of e, when n is an Odd Positive Integer. + n(n-1)2 i sin (n − 4)0 — ..... n-1 2 (−1) = n(n−1) ... + (n+3) 2isin@ [(4) of Art. 158] (n-1) 1. 128 cos 0=cos 80+8 cos 60+28 cos 40+56 cos 20+35. 2. 64 cos 0= cos 70+ 7 cos 50+ 21 cos 30+ 35 cos 0. which are called the exponential values* of the cosine and sine. Cor. From these exponential values we may deduce similar values for the other trigonometric functions. Thus, * Called also Euler's equations, after Euler, their discoverer. (3) Sch. These results may be applied to prove any general formula in elementary Trigonometry, and are of great importance in the Higher Mathematics. Prove the following, by the exponential values of the sine and cosine. Rem. If we omit the i from the exponential values of the sine, cosine, and tangent of e, the results are called respectively the hyperbolic sine, cosine, and tangent of 0, and are written sinh 0, cosh 0, and tanh 0, respectively. Thus we have sinh 0 =−isin i0, cosh 0 = cos 10, tanh 0 =—itan i. Hyperbolic functions are so called, because they have geometric relations with the equilateral hyperbola analogous to those between the circular functions and the circle. A consideration of hyperbolic functions is clearly beyond the limits of this treatise. For an excellent discussion of such functions, the student is referred to such works as Casey's Trigonometry, Hobson's Trigonometry, Lock's Higher Trigonometry, etc. |