the measure of the contained angle, whatever be the radius of the circle, since the arcs are proportional to their radii. Thus, the arc AB or A'B', is the measure of the angle ACB, and is expressed in degrees, &c. B B KA F 22. The complement of an arc is its difference from a quadrant, its supplement, its difference from a semicircle, and its explement, its defect from the whole circumference. Thus if AB be any arc, then BD is the complement, BE the supplement, and BDEFA the explement. The same thing holds with regard to the angles of which the arcs are the measures, that is, if ACB be any angle, BCD its difference from a right angle is called the complement, BCE the supplement to two right angles, and BCA measured by the arc BDEFA, the explement or difference from four right angles. 23. The sine of an arc, or of an angle, of which the arc is the measure, is a perpendicular let fall from one of its extremities upon a radius or diameter passing through the other. 24. The versed sine or versine of an arc is that part of the diameter intercepted between its sine and the circumference. 25. The tangent of an arc is a perpendicular to the extremity of the radius at one end of the arc, and limited by a straight line drawn from the centre, passing through the other. 26. The secant of an arc is the straight line drawn from the centre to the extremity of the tangent. 27. It is usual to express the sine, tangent, and secant of the complement of an arc by the abbreviated terms cosine, cotangent, and cosecant. 28. Let ACDE be a circle of which the diameters AD and CE are at right angles to one another. K CI H B Take any arc AB, produce the radius OB, and draw BG, AK perpendicular to AO or AD, and HB, CI perpendicular to CE; then BG is the D sine, BH or GO the cosine, AG the versine, CH the coversine, DG the suversine, and HE the sucoversine of the arc AB. Also of that arc AK is the tangent, CI the cotangent, OK the secant, and OI the cosecant. E G 29. Since the diameter which bisects an arc, also bisects the chord of that are at right angles, therefore the sine of the arc is equal to half the chord of twice the arc. Thus BG=BF=half the chord of the arc BAF, the double of the arc AB. 30. In the right-angled triangle OGB, BG2+OG2=OB2, that is, the squares of the sine and cosine are together equal to the square of the radius. 31. The triangle OGB being similar to OAK, OG: GB::OA: AK, or the cosine of an arc is to the sine as radius is to the tangent. 32. Also the triangles OGB, OAK being similar as before, OG: OB:: OA: OK, the radius is a mean proportional between the cosine and the secant. 33. Since DG : GB : : GB : GA, it follows that the sine is a mean proportional between the versine and the suversine. 34. Again, AD: AB:: AB: AG, or the chord of an arc is a mean proportional between the diameter and versine. Cor. Since AB2=AD. AG, then because AD is constant, AB2 varies as AG, or († AB)o∞ AG,that is, the square of the sine varies directly as the versine of twice the arc, or inversely as the cosine of twice the arc. 35. The triangles OAK and ICO are similar, therefore AK: AO:: OC: CI; consequently the radius is a mean proportional between the tangent and the cotangent of an arc. CI B H A 36. In the application of algebra to geometry, where the trigonometrical lines are employed, it is necessary to trace their changes in the several quadrants of the circle, since it is obvious that the same lines treated of above, may be applied to each. K In the first quadrant AC, if the sine BG and cosine GO be supposed positive; then the sine B'G' on the same side of the diameter AA', and in the same direction, still remains positive; but the cosine OG' having changed its position with respect to the centre O, or diameter CC', becomes negative. In the third quadrant, the cosine OG' and sine B GB", having both changed their positions, are both negative. In the fourth quadrant, the cosine having resumed its original position, OG is now positive, while the sine GB"", remaining as in the third quadrant, is negative. The tangents and secants depending upon the sines and cosines, have their signs determined accordingly." K R H K From article 30, to 35 and inclusive, R being radius, &c. we obtain * In the above wood-cut, B'"' has been omitted near I'", which may easily be sup plied by the pen. sin 37. Now, since (7) tan.= then it follows from the principles COS of algebra, that when the signs of the sine and cosine are like, the sign of the tangent is positive, and when unlike, the sign of the tangent is negative. In like manner, the signs of the cotangent, secant, and cosecant, may be determined from formulas (8), (9), and (10). Of the Multiples and Powers of Arcs. 38. In most treatises on geometry, such as Leslie's, Legendre's, &c. the elementary propositions containing the principles of trigonometry are also given. It is therefore unnecessary to repeat them here, as it only puts the student to the expense of purchasing the same things in two or three different works. We shall merely give a few of the results most generally useful, referring to those works on geometry and trigonometry where the requisite information may be obtained.* If a and b are two given arcs of a circle of which the radius is unity, then sin (a+b)=sin a cos b+sin b cos a (1) (4) ཙྪཱ If we divide these equations, the one by the other in succession, that is, (1) by (2), and (3) by (4), then cos a cos b+ sin b sin a (5) (6) Dividing the two terms of the second numbers by cos a cos b, and substituting tan a and tan b for their values in terms of the sine and cosine 1+tan a tan b (7) (8) expressions which give the tangent of the sum and of the difference of two arcs in terms of the tangents of these arcs. If we make a=b in the preceding formulæ, they give sin 2 a 2 sin a cos a, cos 2 a=cos2 a-sin2 a (9) (10) (11) * Those we would more particularly recommend are the treatises of Gregory, Woodhouse, Lardner, and Cagnoli. Dr Kelly's Spherics is a very good treatise for teaching the practice of the stereographic projection of spherical triangles. expressions which give the sine, cosine, and tangent of twice the sin (a+b)+sin (a—b) = 2 sin a cos b (12) (13) (14) (15) Let (a+b)=u, and (a—b) —v, then by addition and subtraction a=} (u+v), b= (u-v), consequently the preceding formulæ become sin u+ sin v = 2 sin † (u+v) cos ↓ (u—v) (u—v) cos (u+v) (19) expressions which serve to transform the sum or the difference of the sine or cosine into the product, and thus to unite the two terms into one. If we divide formula (16) by formula (17) they give sin u+ sin v tan (u+v) = sin u- sin v tan (u-v) (20) If we multiply these equations member by member, observing to substitute sin 2 a=2 sin. à cos a, formula (9), then sin 2u—sin2 v — sin (u+v) cos (u+v) (21) (21) Since sin 2 a 2 sin a cos a, and cos 2 a=cos2 a-sin2 a. The second of these equations may be put under the two following forms: = cos 2 a 1-2 sin2 a, and cos 2 a=2 cos2a-1 whence sin a= 1-cos 2 a These expressions are used when, for the squares of the sine and cosine, the first power of the cosine of the double arc is substituted. 40. Let 2a=u, then a=ļu formula (22), these formulæ become 1+tan2 u (24) (25) If b in formulæ (1), (2) be made 2 a, 3 a, &c. we may obtain multiple arcs thus: sin 3 a sin a cos 2 a+ sin 2 a cos a cos 3 a=cos a cos 2 a-sin a sin 2 a Substituting for sin 2 a and cos 2 a, their values, they become power of The coefficients of the different terms are those of the nth the binomial, whence these series may be collected under the following form: sin n a 1 2-1 1 cos at√ sin a cos a-√] sin a 2, a} "(30) cos n a=1{cos á+√—1 sin a}"+} {cos a—√1 sin a}” (31) These formulæ, by development, will give the two foregoing series, and are thus easily verified. 41. It may be shown* that if a represent any arc 1.2.3.4.5.6+, &c. (32) (33) In these expressions the arc a is supposed to be divided by the radius, which is here taken for the unit of length, and consequently a sin if we wish to restore it we must write in place of a and r instead of sin a in the two members of these equations. a a3 2a5 17 at tan a=a+ 1.3*1.3.5+ 1.3.5 3.3.5.7 +&c. 1 a að 2a5 cot a= d&c. versin a= a 3 &c. 2.3.42.3.4.5.6 sin a 1.3. sin a a=sin a+ + + &c. 2.3 2.4.5 (34) (35) (36) (37) (38) These formulæ might be carried much farther than can be introduced into this place. Most of them may be seen by consulting the books already referred to, but, above all, the Analysis Infinitorum of Euler. Tables of Multiples and Powers of Arcs. * Woodhouse's Trigonometry, third edition, page 245,-Gregory, page 42 and 50,-and Cagnoli's copious Treatise. |