As radius 1 AB Calculation. 162: 13333284 nat. tanBC. gent of 53° 07′ 48" : 215.9992 1 : = And, as the fame A 162; 1.6666628 nat. fecant of 269.99937 AC. circumference of a circle whofe diameter is 1; a, b, c being the halves of the three fides of any triangle, and r the radius of its circumfcribing circle. Corollary 6. Since, by prob. 3, b: a + c :: a − c : aa - CC b half the difference of the fegments of the base (6) made by a perpendicular demitted from its oppofite angle, and aa-cc aa+bb. -CC + b we shall have b the fegment adjoining to the fide 2a, for the value of the faid perpendicular to the base, and hence Having now found the value of r, we can calculate all the cafes of trigonometry without any tables, and without reducing oblique triangles to right-angled ones; for having any three parts (except the three angles) given, we can find the rest from thefe five equations: Inftrumentally. The extent from 45° to 53° 08′, upon the tangents, will reach from 162 to 216 upon the numbers. Note. And for the more convenience we may add the three following, which are derived from the 2d, 3d, and 4th, by reverfion of feries.] Suppofe we take here the first example in prob. 1, in which are given two fides 2b = 345, 2c=232, and the angle oppofite to 2c= 37° 20′ = 37} degrees = c. 232 ==== 116=rx: ·651589587 - 04610744 + 00097879 2 000009894 + 000000058 &c =rx (652568435 046117334)=6064511r. Hence r= 116 •6064511 = '9018346. 2 X 116 =191.27677, Again B = 57.2957795 × 1*12402 (the fum of the feries in the third equation)=64 4016 degrees 64° 24'. = = - And A 180-37644016 180 101.735=78.265° 78° 16′ nearly. 78.265 57°2957795 from the fifth equation we have a n =1365982, and r=191*27677, = 191*27677 × 1365982 0017607 +0000288 *0000005= And hence 2a = 374'55368 the third fide of the triangle. Corollary Note. It is common to add another method for right-angled triangles, which is this. ABC being the triangle, make a leg AB radius, that is, with center A and radius AB, defcribe an arc BF then it is evident that the other leg BC reprefents the tangent, and the hypotenufe AC the fecant of the angle A or arc BF. In Corollary 7. As the feries by which an angle is found, often converges very flowly, I have inferted the following approxi mation of it, viz. rr 3r ) nearly; where the letters denote the fame quantities as in the above feries. r 2.3r3 2.4.575 we shall have, by taking the former of thefe from the latter, a3 13a5 12 a 3r 243 640rs = a3 2473 from the former, + &c. But, from the first feries, &c; hence, by fubtracting the latter 3845 A = n × (P 3r n Corollary 8. And again, fince 115 × (P − 2 — § 23) = I 48095 &c; (where q is =), by subtracting this from q &c, and reducing, there will be obtained A = 105 In like manner, if the leg вe be made radius; then the leg AB will reprefent the tangent, and AC the fecant of the arc BG, or of the angle c. But 22 105 - n (144 P-399 – £93) = — — — 3 × (144√ 2 — 2√/1 — q32 — 399 — { q3 which will commonly give the angle true to within a minute of the truth. Where note that the conftant quantity 54567409. And from the whole may be drawn the following problem. 105 = PROBLEM. To perform all the Cafes of Trigonometry without any Tables. HAVING any three parts of a triangle given, except the three angles, the other three parts may be found by fome of the fol lowing fix general theorems. (a+b+c)x(a+b−c) x (a−b+c)x(−a+b+c) But if the hypotenufe be made radius; then each leg will represent the fine of its oppofite angle; namely, the leg AB the fine of the arc AE or angle c, and the leg BC the fine of the arc CD or angle A. And then the general rule for all these cafes, is this, namely, that the fides bear to each other the fame proportion, as the parts which they reprefent. And this is called, making every fide radius, SECT. III. OF HEIGHTS AND DISTANCES, &c. Y the menfuration and protraction of lines and angles, we determine the lengths, heights, depths, or distances of bodies and objects. And this branch Where a, b, c, are the halves of the three fides of the triangle, and A the number of degrees in the angle oppofite the fide 2a, and c the degrees in the angle oppofite the fide 2c; alfor is the radius of the circumfcribed circle; Thus, if the three fides be given, as for example a = 13, 6 = 14, c=15. Then is r = 164, and the angles by these theorems come out as follows, viz. Angles by the Theor. 67 19 179 54 The true Angles. |
