cos (n · 1) B+ Adding and subtracting the two forms of (1) we have, = 2 cos nẞ sin ẞ sin (n + 1) ẞ + sin (n − 1) ß = 2 sin nẞ cos B sin (n + 1) B sin (n − 1) B Similarly from equation (2) we get, cos (n + 1) ẞ= 2 cos nẞ cos B = ẞ, or n = sin 2a = 2 sin a cos 2a = cos2 a But (8) may be changed into (1 — sin2 a) — sin2 a or cos2 a hence, 2 sin a; whence sin2 a = cos 2a 2 cos2 a - 1; whence cos2 a = Then a + ẞ= 2a; and we get, cos a ...... (7) And the same holds good if for a we write a in the three last formulæ, giving, But we may proceed differently, and get another pair of useful forms for tan a; thus, put ß = fa, then a — B = a. cos (a La) = cos a cos a cosa + sin a sin (1 cos a) cos a = sin ja sin a, or tan ja Two other formulæ may be thus obtained which sin2 a + cos2 a = 1, and 2 sin a cos a = sin 2a. traction, - cos a sin a are often useful. We have Hence, by addition and sub sina + 2 sin a cos a + cos2 a = 1 + sin 2a. sin2 a 2 sin a cos a + cos2 a = 1 sin 2a. Extracting the roots, and adding and subtracting the results, cos a = }} {√1 + sin 2a + √I Many other formulæ respecting double and half arcs are easily obtained; but as they are not of frequent use in elementary study, they are left to the student's choice to pursue or not. 3. Multiple arcs are generally most elegantly expanded by Demoivre's theorem, hereafter to be given; but as for small multiples, they frequently occur in early stages of trigonometry. One method, that of successive deduction, is indicated, rather by example than precept, but sufficient for the present purpose. It will be kept in mind that sin 2a and cos 2a have been found in (7, 8). Then sin 3a sin (2a + a) = sin 2a cos a + cos 2a sin a 4 sin3 a. 3 cos a + 4 cos3 a. = 2 sin a (1 sin2 a) + sin a (1 - 2 sin2 a) = 3 sin a cos 3a cos (2 a + a) = cos a cos 2a sin 2a sin a=Similarly sin 4a = sin (3a + a), cos 4a = cos (3a + a), and so on. V. THE EXPRESSIONS FOR SIN (a ± ß) AND COS (a ± ß) WHEN а IS SOME WHOLE NUMBER OF QUADRANTS. and for all additions of quadrants these values will be repeated in the same order. The same results might have been inferred by combining the four figures at p. 422 in one, and reasoning from known geometrical relations. The tangents and co-tangents might also be inferred from the figure in the same way; or they might be obtained from these results by the equation and cot y = COS Y ciprocals of the cosine and sine have the same signs as those functions. Some interesting discussions of the signs connected with these functions may be found in Professor Young's Mathematical Dissertations, p. 8. VI. THE VALUES OF THE TRIGONOMETRICAL FUNCTIONS OF CERTAIN ARCS. 1. To find cos 30°, sin 30°, and tan 30°. If a 30° we have cos 3a = cos 90°0. Whence taking the value of cos 3a from p. 428, we have, 0. Whence cos 30°, or cos 30= 4 cos3 30° 3 cos 30° cos 90° cos2 30° sin n° 1 = = √3. cos 30° √3 1 2. Find sin 45°, cos 45°, and tan 45°. √/3, =; or sin 30°. Whence also As before, cos 2.45° = cos 90° = 0; and hence 2 cos2 45°-1=0; or cos2 45°, and sin2 45° = 1-cos2 45° = }; and we have cos 45° 2, sin 45° = √2, and tan 45° = 1. = 3. Find sin 60°, cos 60°, tan 60°. These may be inferred from (1), since the sine, cosine, and tangent of 30°, are the cosine, sine, and cotangent of 60°; but for illustration of the method of proceeding, the investigation is annexed independently of 30°. sin 3.60° sin 180° = 0; whence, as before, 3 sin 60° 4 sin3 60° 0, or sin2 60° = , cos2 60° = 1 − 1 = }; and sin 60° = }} √3, cos 60° and tan 60° = √3. -- = Having obtained the functions of these arcs, (the final expressions for which are the simplest that occur throughout the quadrant, for any arcs,) we can continually obtain their halves or doubles: their halves by the resolution of quadratic equations, and their doubles by squaring certain functions of the sine and cosines already obtained. We can also obtain expressions containing the functions of the third part of an arc by the resolution of cubic equations, and so on to any extent. Two examples are annexed, to find the functions of 15° and 10°, the half and the third parts of 30°. cos 30° = cos 2.15° 2 cos2 15° − 1 = } √ 3 ; hence cos2 15° = (2+3), and sin2 15° = 1 cos 15° (2 Hence, extracting the roots, we have the following expressions of value: cos 150 = {√ √}, sin 15° = } { √ } − √ }}, tan 15° = 2 — √3. Again, for sin 10° we have sin 3.10° = - 4 sin3 100+ 3 sin 10°: hence 4 sin3 10° 3 sin 10° 0, and by Cardan's formula, we have, This expression taking an imaginary form, indicates that all three roots are real, whilst neither of them can be exhibited in a real form by such a process. The same circumstance happens universally in obtaining the sine or cosine of an arc, by supposing it a third part of an arc whose sine or cosine are given, except when that given sine or cosine is 0. The method of trisection, therefore, is inapplicable to the finding of useful expressions * for these functions; but it gives an opportunity of making a remark which will be further expanded in the second volume. We have sin 30° = sin 390° = sin 750° = 1; and hence the problem which is virtually put into equations, has in reality three different cases, according as we suppose these three angles to be trisected. Hence the roots are sin 10°, sin 130°, and sin 250°, all which are real, and answer to the real roots of the equation before found. We might, hence, have anticipated this result: and, indeed, the However, in all cases the values of the roots can be readily calculated by Horner's method; and as the same reasoning will apply to every section of a given arc, it is quite clear that we can always actually compute any function of any given part of an arc or its angle, when we are in possession of the value of any one of its trigonometrical functions. double values of the radicals in the solution of the other problems indicate the same kind of circumstance, viz. two values of the sine, cosine, and tangent sought; which, on the same principle, were indications of the sine, cosine, and tangent of a and π + 1⁄2 a. The surd values of the sines, and hence of the cosines, and the tangents which may be obtained from them, are given in the Introduction to Hutton's Tables, p. xxxix. for every third degree of the quadrant. The deduction of these will furnish sufficient exercise to the student. VII. THE CALCULATION OF TRIGONOMETRICAL FUNCTIONS. THESE functions can be expressed in a series of positive integer powers of the arc itself, and the coefficients of the series determined; and conversely, the arc can be expressed in a series of positive integer powers of any one of these functions. These series may be found either by indeterminate coefficients, or by the differential and integral calculus. The former method, however, is laborious; and the latter implies a degree of acquirement beyond our present progress. Hence, we shall adopt a more simple method of proceeding in this place, leaving the deduction of the series in question for its proper analytical position in the Course. The method is founded on the principle, that in very small arcs the sine varies very nearly as the arc itself. For let a be a minute arc, and ẞ one more minute, by which a is increased. Then sin (a + ẞ) sin a cos ẞ + cos a sin ẞ; and since a and B are minute arcs, cos a and cos ẞ are very nearly equal to unity. Hence, taking them actually as unity, we get sin (a + ß) = sin a + sin ß, and the arc, therefore, increases nearly as the sine, when these arcs are very small. Now, by IV. 15, we have sin a= {√1 + sin 2a If, then, we put 2a successively equal to 30°, 30° 30° √1- sin 2a}. 30° 2 22 211 and compute The cosine, tangent, or any other function of l′ can now be obtained, as cos 1'=√1 sin 21′ = 9999999577, and tan 1′ = whence so far as the first ten decimals, there is no difference between the sine and tangent of 1'. - Again, from (IV. 3,) we have sin (n + 1) ß = 2 sin nẞ cos ß sin (n - 1) ß: and if we put n = 1, 2, 3, ... 1799, and ẞ = 1', we shall be able to calculate the sines of all angles from 0° to 30°, for every minute of a degree; and, concurrently, all the other trigonometrical functions of the same arcs. To calculate those from 30° to 45°, we may use the formula thus obtained :sin (30° + ẞ) + sin (30°— (ß) = 2 sin 30° cos ẞ = cos ẞ, from which, sin (30° +6) = cos ẞ — sin (30°— B) and making ẞ successively equal to 1′, 2′, 3′, .... 899', we shall obtain the sines, and thence the other functions of the arcs from 30° to 45° inclusively. Also, the sine of any arc is the cosine of its complement; and hence, as we have computed all the complementary functions, we have the direct functions of all arcs from 45° to 90°, and the functions of the entire quadrant are computed. The functions of arcs greater than 90° are at once obtained from the equations chapter V., p. 428. VIII. THE CONSTRUCTION AND USE OF THE TABLES OF TRIGO NOMETRICAL FUNCTIONS. 1. SINCE by the preceding method we can calculate the sines to radius 1, of all the angles from l' up to 90°, we may suppose them prepared for tabulation; and thence also by means of the relations deduced in chapter II. all the other functions. In Hutton's tables they are computed to seven decimal places. On each page are given the values of all the functions, sin, cos, tan, cot, sec, cosec, versin, and coversin, of all the minutes from p degrees to (p + 1) degrees inclusive. The number p, if under 45°, is found at the head of the page to the left; and if 45° or upwards to 90°, at the bottom of the page to the right. The minutes, if p be less than 45°, are numbered from the top 0', to 60' at the bottom, the numbers being the left column of the page; but if 45° or upwards, they range from the bottom o' to the top 60', and constitute the right column of the page. The name of each column of functions is placed at the top or bottom as p is less or greater than 45°. = It will also appear that the degrees at the top and the minutes at the left side, together with the degrees at the bottom and the minutes at the right side, of any horizontal column, together make 90°; or in other words, that any given function of a given arc is the complementary function of the complement of that arc in the structure of the tables. Thus sin 9° 10' 1593069 = cos 80° 50' (see page 286 of the Tables), and so of the other functions. This is an arrangement depending on the equation sin a = cos (90 — a), and reduces the table to half the dimensions it would otherwise require to carry the functions up to 90°. These natural sines, natural cosines, etc. are always placed on the left page whenever we open the tables, and headed "NATURAL SINES, &c." The differences between the sines and between the cosines of each two consecutive arcs differing by 1', are placed in columns and adjacent to them, marked " DIFFERENCES;" thus sin 9° 10' sin 9° 9' 1593069-15901970002872, the effective figures 2872 of which is found on a line lying horizontally between sin 9° 9′ and 9° 10'; or again cos 9° 9′ — cos 9° 10′ = ·9871827 — ·9871363 = *0000464, and the effective figures 464 are put down horizontally. = Again, let a be any number of degrees and minutes; then since covers a = 1 sin a, we have covers a― covers (a + 1') = (1 − sin a) — {1 − sin (a+1')} =sin (a + 1') — sin a. The differences between two consecutive coversines is equal to the difference between the sines of the same angles. The coversines of the angles are therefore put down on the opposite side of the column of differences from the sines, the same difference applying to each of the columns. For the same reason the versines are placed on the opposite side of the column of differences from the cosines. No other remark remains to be made on the table of natural functions. The table of "LOG SINES, &c." on the right hand page is formed by taking the logarithms of the numbers on the opposite page. Thus log sin 9° 10′ = log 1593069 = 1.2022345, and so of all the rest. However, to avoid the negative indices in the logarithms, which would create great difficulties in printing and much liability to mistakes in calculation, 10 is added to all the logarithms of the sines, etc. throughout the entire tables. Hence tabular sin a = 10+ log sin a, and hence tab. sin 9° 10′ = 10 + 1·2022345 = 9·2022345; and similarly with all the other functions and values of a. The succession of columns in the two tables is different. In the table of |