3. Since arcs are proper and adequate measures of plane angles, (the ratio of any two plane angles being constantly equal to the ratio of the two arcs of any circle whose centre is the angular point, and which are intercepted by the lines whose inclinations form the angle), it is usual, and it is perfectly safe, to apply the above names without circumlocution as though they referred to the angles themselves; thus, when we speak of the sine, tangent, or secant, of an angle, we mean the sine, tangent, or secant, of the arc which measures that angle; the radius of the circle employed being known. 4. It has been shown in the 1st vol. (pa. 382), that the tangent is a fourth proportional to the cosine, sine, and radius; the secant, a third proportional to the cosine and radius; the cotangent, a fourth proportional to the sine, cosine, and radius; and the cosecant a third proportional to the sine and radius: Hence, making use of the obvious abbreviations, and converting the analogies into equations, we have Or, assuming unity for the rad. of the circle, these will become These preliminaries being borne in mind, the student may pursue his investigations. b 5. Let ABC be any plane triangle, of which the side BC opposite the angle A is denoted by the small letter a, the side AC opposite the angle в by the small letter b, and the side AB opposite the angle c by. the small letter c, and CD perpendicular to AB: then is, c=a. cos B + b. cos A. D A C B For, since Acb, AD is the cosine of a to that radius; consequently, supposing radius to be unity, we have AD = 6, COS A In like manner it is BD a. cOS. B. Therefore, AD + BD = AB c a. cos. B+b. cos A. By pursuing similar reasoning with respect to the other two sides of the triangle, exactly analogous results will be obtained. Placed together, they will be as below: 6. Now, if from these equations it were required to find expressions for the angles of a plane triangle, when the sides are are given; we have only to multiply the first of these equations by a, the second by b, the third by c, and to subtract each of the equations thus obtained from the sum of the other two. For thus we shall have b2 + c — a2 =2bc. cos. A, whence cos. A = a2 + c2 — b2 = 2ac. cos. B, ... COS. B, = >(II.) 2ac 7. More convenient expressions than these will be deduced hereafter: but even these will often be found very convenient, when the sides of triangles are expressed in integers, and tables of sines and tangents, as well as a table of squares, (like that in our first voi.) are at hand.) Suppose, for example, the sides of the triangle are a = 320, b = 562, c = 800, being the numbers given in prop. 4, pa. 161, of the Introduction to the Mathematical Tables: then we have b2+c2 —a2 = 853444 2bc log. 5.9311751 The remainder being log. cos. A, or of 18°20′ = 9.9773671 = 899200 The remainder being log. cos. B, or of 33°35′ 5.7092700 = 9.9207060 Then 180°. (18° 20'+33° 35′) = 128° 5′ = c : where all the three triangles are determined in 7 lines. 8. If it were wished to get expressions for the sines, instead of the cosines, of the angles; it would merely be necessary to introduce into the preceding equations (marked II), V(1 instead of cos. A, COS. B, &c. their equivalents cos. A = sin, A), cos. B = √ (1 - sin2. B), &c. For then, after a little reduction, there would result, sin. A 1 2bc 1 ·√2a2b2 + 2a2c2 + 2b2c2 — (aa+b*+c*) sin. B = zac√2a2b2 + 2a2c2 + 262c2 — (a^+b^+c3) 1 sin. c = ab2a2b2 + 2u2c2 + 262c2 — {a^+ b*+c*) Or, resolving the expression under the radical into its four constituent factors, substituting s for a+b+c, and reducing, the equations will become. These equations are moderately well suited for computation in their latter form; they are also perfectly symmetrical: and as indeed the quantities under the radical are identical, and are constituted of known terms, they may be represented by the same character; suppose K: then shall we have Hence we may immediately deduce a very important theorem: for, the first of these equations, divided by the second sin. A sin. B: sin. c α a:b; c ... (IV.) Or, in words, the sides of plane triangles are proportional to the sines of their opposite angles. (See th. 1, Trig. vol. i). 9. Before the remainder of the theorems, necessary in the solution of plane triangles, are investigated, the fundamental proposition in the theory of sines, &c. must be deduced, and the method explained by which Tables of these quantities, confined within the limits of the quadrant, are made to extend to the whole circle, or to any number of quadrants whatever. In order to this, expressions must be first obtained for the sines, cosine, &c. of the sums and differences of any two arcs or angles Now, it has been found (I) that And the equations (IV) give a = b. cos. c + c. cos. B sin. c .c=a. sin. A Substituting these values of b and c for them in the preceding equation, and multiplying sin. A = sin. B. cos. c + sin. c. cos. B. But, in every plane triangle, the sum of the three angles is two right angles; therefore B and c are equal to the supplement of A: and, consequently, since an angle and its supplement have the same sine (cor. 1, pa. 378, vol. i), we have sin. (B +c) = sin. B. cos. c+ sin c. cos.. 10. If, 10 If, in the last equation, c become subtractive, then would sin. c manifestly become subtractive also, while the cosine of c would not change its sign, since it would still con tinue to be estimated on the same radius in the same direction. Hence the preceding equation would become. sin. (B-C) sin B Cos. c sin. c. cos B. = 11. Let c' be the complement of c, and be the quarter of the circumference: then will c'O- -c, sin. ccos. C, and cos. c' sin c. But (art. 10), sin. (B c') = sin. B, cos. C' sin. c' cos B. Therefore, substituting for sin. c', cos. c', their values, there will result sin (B— C') = sin. B. B. COS. C. But because c' = 10 c. we have sin. (BCO)=sin. [(B+ c)] = C) 10] (B+ c)]= COS (B+C). Substituting this value of sin (B — c') in the equation above, it becomes cos. (B+C)= COS B COS. C. sin. B sin c. COS sin. c sin. (BC) sin 12 In this latter equation, if c be made subtractive, sin. c will become sin c, while cos. c will not change: conse. quently the equation will be transformed to the following, viz. cos. (BC) COS. B. COS. C + sin. B. sin c. A If, instead of the angles B and C, the angles had been a and в; or, if ▲ and в represented the arcs which measure those angles, the results would evidently be similar: they may therefore be expressed generally by the two following equations, for the sines and cosines of the sums or differences of any two arcs or angles: sin. B. COS. A. 2 } (v.) sin. (AB) = sin. A. COS. B COS. (A + B) = COS. A, COS. B. 13. We are now in a state to trace completely the mutations of the sines, cosines, &c. as they relate to arcs in the various parts of a circle; and thence to perceive that tables which apparently are included within a quadrant, are, in fact, applicable to the whole circle. N Imagine that the radius Mc of the circle, in the marginal figure, coinciding at first with AC, turns about the point c (in the same manner as a rod would turn on a pivot), and thus forming successively with ac all possible angles: the poin: м at its extremity passing over all the points of the circumference ABA B'A, or describing the whole circle. Tracing this motion attentively, it will appear, that at the point A, where the arc is nothing, the sine is nothing also, while the cosine does not differ P B MA M P from the radius. As the radius мc recedes from ac, the sine PM keeps increasing, and the cosine CP decreasing, till the describing point м has passed over a quadrant, and arrived at B in that case PM becomes equal to cв the radius, and the cosine CP vanishes. The point м continuing its motion beyond B, the sine P'M' will diminish, while the cosine cp', which now falls on the contrary side of the centre c will increase. In the figure, p' M' and CP' are respectively the sine and cosine of the arc a'm' or the sine and cosine of ABM', which is the supplement of A'M' to O, half the circumferen ce: whence it follows that an obtuce angle (measured by an arc greater than a quadrant) has the same sine and cosine as its supplement; the cosine however, being reckoned subtractive or negative, because it is situated contrariwise with regard to the centre c. When the describing point м has passed over O, or half the circumference, and has arrived at A', the sine P'M' vanishes, or becomes nothing, as at the point A, and the cosine is again equal to the radius of the circle Here the angle ACM has attained its maximum limit; but the radius cм may still be supposed to continue its motion, and pass below the diameter AA. The sine, which will then be P"M", will consequently fall below the diameter, and will augment as M moves along the third quadrant, while on the contrary cp", the cosine, will diminish In this quadrant too, both sine and cosine must be considered as negative; the former being on a contrary side of the diameter, the latter a contrary side of the centre, to what each was respectively in the first quadrant. At the point B', where the arc is three-fourths of the circumference, or, the sine P"M" becomes equal to the radius CB, and the cosme cp"vanishes. Finally, in the fourth quadrant, from B'to A, the sine p""M"", always below ▲▲', diminishes in its progress, while the cosine cp", which is then found on the same side of the centre as it was in the first quadrant, augments till it becomes equal to the radius CA. Hence, the sine in this quadrant is to be considered as negative or subtractive, the cosine as positive. If the motion of м were continued through the circumference again, the circumstances would be exactly the same in the fifth quadrant as in the first, in the sixth as in the second, in the seventh as in the third, in the eighth as in the fourth and the like would be the case in any subsequent revolutions. 14. If the mutations of the tangent be traced in like manner, it will be seen that its magnitude passes from nothing to infinity in the first quadrant; becomes negative, and decreases from infinity to nothing in the second; becomes positive again, and increases from nothing to infinity in the third |