+ α, and sin α = cos a a; consequently by (13, 14, 28, 29, 32, 33) of Art. 15, = sin ( cos (w + α) These relations will also hold, if we add to each arc, or take from it, for the resulting arcs will have the Whence, adding 2 n to each in (1), any number of circumferences; same extremities as at present. By taking away 2 n w from each arc, we get, in a similar way, (3). (4). we have Hence if p be a positive or negative integer of the series, 0, 2, 4, and a positive or negative integer of the series 1, 3, 5 by (3) and (4), sin a. (8). cos α . = cos a. 30. To develop the sine ascending powers of x. Let (9), cos (p α) = (11), cos (qw and cosine of an angle x in a series of sin x = ee x + €2 x2 + €3 x3 + etc. .... then by changing x intox, and remembering (Art. 15) that sin (- x) sin x, and cos (− x) = cos x, we obtain sin x = ex + e3 x3 + €5 x3 + etc. cos x = Co + C2 x2 + C1 x2 + cx + etc. Again, by (23, 25) of Art. 16, we have the relations half the sum of (2) sin z = 2 cos(x + z) sin ‡ (x − z), COS Z COS X = 2 sin(x+2) sin † (x − z), which by (5) and (6), become Dividing both sides of each of these by x- z, and taking z = x in the results, we get and by substituting in these the values of sin x and cos x given in (5) and (6), we obtain e (Co+cq x2 + c1x*+c, x+.. etc.) e+3e, x2 + 5 ez x* +7 e, x® +..etc. e (e x +еz x2+еs 25+ etc...) = 2 c2 x + 4 c, x2 + 6 cx3+ etc... Since these two equations are identical, we have (Algebra, Art. 135), e = eco, 3 e3 = e cg, 5 es = ec11, 7e = e c6, etc., — e3, 4 CA = 2' 1.2.3 1.2.3.4' es 1.2.3.4.5 Substituting these values in (5) and (6), we get 1 e 1.2.3 90° is the unit of circular measure, as in Art. 4. If, then, we take = 1, ore 1, the equations (7) and (8) become in which it is to be understood that x is an arc of the circle to radius unity. The sine and cosine of an angle A' will be obtained from (9) and (10), by writing for x in the right-hand members of these equations This elegant method of deducing sin x and cos x is due to Mr. W. Finlay, Professor of Mathematics at Manchester.-(Mathematician, page 299, vol. iii.) sin x Cor. Since tan x = we see by the preceding expressions for sin x and cos x, that the development of tan x begins with x. Hence assume *As r may be any angle in (7), whatever value is found for e for any value of x, the same value of e will belong to the series in general. In this write the values of sin x and cos x, already given, multiply out, and equate the coefficients of the like powers of x; we then get 2 x3 + 24x3 1.2.3.4.5.6 31. To prove Demoivre's Theorem, that for all values of n, By multiplication, (cos +1 sine) (cos +√1 sin 0) = cos 20 sin 20 ± 1 2 sin cos 0, or (cos +1 sin 0) = cos 20 ±√1 sin 20, by (5, 6,) of Art. 16. 20±√ Also, by multiplication and the preceding, (cos +1 sin e) (cos +1 sin ) (cos± 0) = (cos 20 ±√-1 sin 2 0) (cos e±√ = 1 sin ) sin 20 sin± √ 1 (sin 20 cos +cos 2 0 sin 0), or (cos +1 sin e) cos 3 0 ± 1 sin 3 0, by (15, 16), Art. 16. = 30±√1 Hence when the index n is a positive integer, (cos 1 sin 0)* = cos no ± √ — 1. sin n 0. Next let the index be a negative integer; then because 1 = sin 20+ cos 20, cos 0-1 sin cose-1 sin 0/ √1 sin 0)", by actual division. 1 sin 0)" = cos(no) ±√1 sin ( This proves the theorem for negative indices. (cos e 1 sin 0)" = cos no ± √ 32. In the construction of trigonometrical tables, the numerical expressions of some function (sine or cosine) for certain values of the are or angle, at large intervals, are first determined, as in Art. 15, and tabulated; and then the intervals are filled up by the calculation of those values of the function that belong to the intermediate values of the arc or angle. To secure accuracy, which is of the greatest importance in tables for use, formula of verification, as they are called, are employed, by which the value of any function already computed is again calculated by some independent method. The agreement of the value thus found, with that obtained by the other method, is the test of accuracy. As all the tables referred to in this Article have been computed already to the greatest accuracy, it will be sufficient to describe here, as briefly as possible, the mode of computation usually employed; and first to find sin l'. By (9) of Art. 16 (A being less than 45), Let A sin 15° = sin A= {√ (1 + sin 2 A) (1 sin 2 A) }. = 15; then we have by (36) of Art. 15, {√ (1 + sin 30) (1 − sin 30 )} = } (√ } − √ }). 30° Hence sin 15° or sin is known, and thence cos 15 by (1) of Art. 15.. Now, by noticing the sines of the last of these and its double, it will be found that the sines of small arcs are nearly as the arcs themselves. 302 Hence if sin be denoted by s, 2" arc 52" 734375: arc l's: sin l'. This gives sin l', and thence cos 1' by (1) of Art. 15. The values of sin l' and cos 1', found in this way, are sin l' = .0002908882, cos l' = ⚫9999999577, nearly. The tangent or any other function may now be obtained, as It will be seen by this that the sine and tangent of 1' agree to ten decimal places. The sine of 1' may also be calculated by the formula (Art. 29), sin (n + 1) B = 2 sin n B. cos B — sin (n Hence, if in this we put n = 1, 2, 3,. 1', we shall be able to calculate the sines of all angles from 0 to 302, for every minute of a degree, and consequently all the other trigonometrical functions. After proceeding as far as 30, the labour of computation is considerably reduced by the formula (11), Art. 16. By transposition that formula becomes 30; then remembering (Art. 15) that sin 30° , we have = Consequently the values of the sines of all angles from 30 to 45° may be found by simply taking the differences of previously found values. And similarly for the corresponding values of the cosines, by means of the formula (12), Art. 16. It is unnecessary to carry the operation beyond 45°, as the sines of all angles less than 45° would give the cosines of their complements, etc. By this method, then, the functions of the entire quadrant are computed. The labour of "filling up the intervals" is very much reduced by the method of differences. (See Algebra, Art. 169.) |