SECTION VIII. COMPUTATION OF THE CANON. ART. 223. THE trigonometrical canon is a set of tables containing the sines, cosines, tangents, &c., to every degree and minute of the quadrant. In the computation of these tables, it is common to find, in the first place, the sine and cosine of one minute; and then, by successive additions and multiplications, the sines, cosines, &c., of the larger arcs. For this purpose, it will be proper to begin with an arc, whose sign or cosine is a known portion of the radius. The cosine of 60° is equal to half radius. (Art. 96. Cor.) A formula has been given, (Art. 210,) by which, when the cosine of an arc is known, the cosine of half that arc may be obtained. 30° 150 By successive bisections of 60°, we have the arcs 0° 28' 711 30/11 0 14 3 45 If the radius be 1, and if a=60°, b=30°, c-15°, &c.; then cos b=cos α=√1+1× =0.8660254 Proceeding in this manner, by repeated extractions of the square root, we shall find the cosine of 0° 0′52′′ 44′′|| 3|||| 45′′|| to be 0.99999996732 And the sine (Art. 94.)=v1 COS 2 0.00025566346 This, however, does not give the sine of one minute exactly. The arc is a little less than a minute. But the ratio of very small arcs to each other, is so nearly equal to the ratio of their sines, that one may be taken for the other, without sensible error. Now the circumference of a circle is divided into 21600 parts, for the arc of 1'; and into 24576, for the arc of 0° 0′ 52" 44||| 3|||| 45||||| Therefore, 21600 24576:: 0.00025566346 0.0002908882, which is the sine of 1 minute very nearly.* And the cosine v1-sin2 = 0.9999999577. 224. Having computed the sine and cosine of one minute, we may proceed, in a contrary order, to find the sines and cosines of larger arcs. Making radius =1, and adding the two first equations in art. 208, we have sin (a+b)+sin (a—b)=2 sin a cos b Adding the third and fourth, If we put b=1', and a=1', 2', 3', &c. successively, we shall have expressions for the sines and cosines of a series of arcs increasing regularly by one minute. Thus, *See note H. sin (1/+1)-2 sin 1'xcos 1-sin 0-0.0005817764, &c. &c. cos (1'+1')=2 cos 1'xcos 1'-cos 0=0.9999998308, The constant multiplier here, cos 1' is 0.9999999577, which is equal to 1-0.0000000423. 225. Calculating, in this manner, the sines and cosines from 1 minute up to 30 degrees, we shall have also the sines and cosines from 60° to 90°. For the sines of arcs between 0° and 30°, are the cosines of arcs between 60° and 90°. And the cosines of arcs between 0° and 30°, are the sines of arcs between 60° and 90°. (Art. 104.) 226. For the interval between 30° and 60°, the sines and cosines may be obtained by subtraction merely. As twice the sine of 30° is equal to radius (Art. 96,) by making a= 30°, the equation marked I, in Article 224, will become sin (30°+b)=cos b—sin (30°—b.) And putting b=1', 2', 3', &c., successively, If the sines be calculated from 30° to 60°, the cosines will also be obtained. For the sines of arcs between 30° and 45°, are the cosines of arcs between 45° and 60°. And the sines of arcs between 45° and 60°, are the cosines of arcs between 30° and 45°.* (Art. 96.) 227. By the methods which have here been explained, the natural sines and cosines are found. The logarithms of these, 10 being in each instance added to the index, will be the artificial sines and cosines by which trigonometrical calculations are commonly made. (Arts. 102, 3.) 228. The tangents, cotangents, secants, and cosecants, are easily derived from the sines and cosines. By Art. 93, * See note I. SECTION IX. PARTICULAR SOLUTIONS OF TRIANGLES.* ART. 231. ANY triangle whatever may be solved, by the theorems in Sections III. IV. But there are other methods, by which, in certain circumstances, the calculations are rendered more expeditious, or more accurate results are obtained. The differences in the sines of angles near 90°, and in the cosines of angles near 0°, are so small as to leave an uncertainty of several seconds in the result. The solutions should be varied, so as to avoid finding a very small angle by its cosine, or one near 90° by its sine. The differences in the logarithmic tangents and cotangents are least at 45°, and increase towards each extremity of the quadrant. In no part of it, however, are they very small. In the tables which are carried to 7 places of decimals, the least difference for one second is 42. Any angle may be found within one second, by its tangent, if tables are used which are calculated to seconds. But the differences in the logarithmic sines and tangents, within a few minutes of the beginning of the quadrant, and in cosines and tangents within a few minutes of 90°, though they are very large, are too unequal to allow of an exact determination of their corresponding angles, by taking proportional parts of the differences. Very small angles may be accurately found, from their sines and tangents, by the rules given in a note at the end.t 232. The following formulæ may be applied to right angled triangles, to obtain accurate results, by finding the sine or tangent of half an arc, instead of the whole. In the triangle ABC (Fig. 20, Pl. II.) making AC radius, By conversion, (Alg. 389, 5.) • Simpson's, Woodhouse's, and Cagnoli's Trigonometry. + See note K. |