we have cos x = 1. Therefore, putting x = 0, equation (2) will become 1 = Co which is the required value of c.. Again, because cos (− x) = cos x, the development of cos x must remain unaltered when we change x into x. Because this change will reverse the signs of all the odd powers of x, the coefficients of c,, c,, etc., of these powers must all vanish, and the development must be 7. We must now choose some property of the sine and cosine which will enable us to form equations of condition for the coefficients 8, 8, ,etc., and c2, C, C, etc. The most simple property for this purpose is that expressed by the addition theorem: sin (x + y) sin x cos y + cos x sin y. (5) Because the equation (3) is to be true for all values of x, it must remain true when x+y is substituted for x. Making this substitution in (3), we have sin (x+y) = x + y +8, (x + y)3+s, (x + y)3 + etc. Developing the powers in the second member, and collecting the terms multiplied by the first power of y, we have sin (x+y)=x+8,x2 +8 ̧x2 +8,x2 + ... +y(1+38,x2+58,x* + 78,x° + . . .) which we need not compute. } (6) From (5) we have, by substituting for cos y and sin y their assumed developments, (3) and (4), Now the expressions (6) and (7) must be identically equal; therefore the coefficients of each power of y must be identically equal. Equating the coefficients of the first power of y, we have 138,x2+58,x+78,x+ etc. cos x. But we have, by (4), cos x = 1+c2x2 + cx1 + etc. This equation must be satisfied for all values of x. Equating the coefficients of like powers of x, we find cos (x + y) cos x cos y sin x sin y. We find, by substituting x+y for x in (4), cos (x+y)=1+c,(x + y)2 + c,(x + y)* + etc.; from which, developing to the first power of y as before, (9) (10) From the second member of (9), by substituting for sin y and cos y their developments, namely, we find cos y = 1+ c'y' + etc., sin y = y +8,y3 + etc., cos (x+y)= cos x+y(− sin x) + terms × y3, y3, etc. (11) Equating the coefficients of y in (10) and (11), we have The equations (8) and (12), taken alternately, solve our problem. The law of the coefficients is obvious. Substituting them in the developments (3) and (4), we have 78. Convergence of the series. These series are convergent for all values of x, a result which may be shown thus: The ratio of the successive pairs of adjacent terms in the development of the sine are, omitting the minus sign, x2 2.3' x2 4.5' x2 6.7' x2 8.9' etc.; that is, each term is formed from the preceding one by multiplying by one of these factors. Now, however great may be x, we can continue these fractions so far that their denominators shall become greater than 2, and so their values less than. After this point the sum of all the following terms will be less than that of a geometric progression of which the first term is the term of the development (13), whose quotient by the preceding term is less than, and whose ratio is . Such a progression has a limit, whence the sum of the series (13) must also have a limit. The following two applications of these series will be useful as exercises: 1. Square each series, carrying the square as far as the sixth or eighth power of x, and show that the squares fulfil the condition sina+cosx = 1 identically. 2. Compute the values of the sine and cosine of 10° and 30° to 5 places of decimals, remembering that we are to take the natural measure of the arc a in radians (§ 14), and compare the result with that in the tables. 79. Sine and cosine in terms of imaginary exponentials. It is shown in algebra that if we call e the sum the series, or 1 1 1 1+1+ + + + etc., ad infinitum, Putting i, the imaginary unit, =-1, substituting xix V-1 and reducing, this equation becomes for x, The sum and difference of these equations are 2 cos x = exi+e-xt = exor + 2i sin x = exi - e-xi = exi 1 For some purposes these equations may be written in the symmetric form These are two of the most celebrated equations in algebraic trigonometry, and are called EULER's equations, after their discoverer, LEONHARD EULER. 80. Demoivre's theorem. If in the equation (14) we substitute nx for x it becomes enxi cos nx + i sin nx. enxi = (cos + sin x)". By raising (14) to the nth power we have Therefore (cos x + sin x)” = cos nx + i sin nx, which is known as DeмOIVRE's theorem. (16) This theorem enables us to develop the sines and cosines of multiples of an angle in powers of the sine or cosine of the simple angle, as follows: 81. PROBLEM. To develop sin nx and cos nx in powers of sin x and cos x. S Developing the first member of (16) by the binomial theorem, and substituting for the powers of i their values (Algebra, § 325), namely, i = - 1, 23 =- -i, i=+1, This development being identically equal to the second member of (16), we have, by equating the real terms and putting, for brevity, c = cos x, 8 = sin x, This series will go on to infinity unless n is a positive integer, * We here use the very convenient abbreviated notation for the binomial co |