. Elements of plane and spherical trigonometry . le, Fig. 21, BD = AB — AD, and in the second, BD = AD — AB;but in both cases BD2 = AB2 + AD2 — 2AB . both sides of this equation add CD2, then BD2 + CD2 = AB2 + AD2 + CD2 — 2AB . AD;but BD2 + CD2 = BC2, AD2 -f CD2 = AC2,and AD = AC . cos A : ^C2 = AB2 + ZC2 — . ZC. cos .A,or a2 = b2 -f- c2 — 26c cos In both of these cases the side a is opposite an acute prove the same thing for a side opposite to an obtuseangle, we have, in the second triangle, Fig. 21, AD = AB + BD: AD2 = AB2-\- BD2 -f 2AB . BD. 74 PLANE TRIGONOMETRY. Ad
. Elements of plane and spherical trigonometry . le, Fig. 21, BD = AB — AD, and in the second, BD = AD — AB;but in both cases BD2 = AB2 + AD2 — 2AB . both sides of this equation add CD2, then BD2 + CD2 = AB2 + AD2 + CD2 — 2AB . AD;but BD2 + CD2 = BC2, AD2 -f CD2 = AC2,and AD = AC . cos A : ^C2 = AB2 + ZC2 — . ZC. cos .A,or a2 = b2 -f- c2 — 26c cos In both of these cases the side a is opposite an acute prove the same thing for a side opposite to an obtuseangle, we have, in the second triangle, Fig. 21, AD = AB + BD: AD2 = AB2-\- BD2 -f 2AB . BD. 74 PLANE TRIGONOMETRY. Adding CD2 to both sides, we Fig. 21. have ZD2 + CD2 = AB2 -f BB2 + CD2 -j- 2A5 . ZD2 + CD2 = b2,BD> Jr(W2 = a2,and 4 BD = a cos BBC= a cos (w — E) = —a cos B;.-. 62 = c2 + a2 — 2ca cos we may write in any tri-angle, a2 = b2 -f c2 — 26c cos ^;fr2 — c2 -j- ^2 — %ca cos i?,c2 =a2+b2 — 2abcos C) 53. To determine the sines of the half-angles of a triangle interms of the sides. From the first of (84) we have. (84) cos A b2-\- & 2bc Subtracting both sides of this equation from unity, we have cos A 2bc__2bc—b2—c2+a22bc 2bc (a-f5-c)(fl-Hc).2bc . (57), • 2 i a (a -\-b — c) (a — b 4- c)sin2 \A = *—L_ LA !—/. 4bc Now, letSubtracting a -j- 6 -f- c = 2s2c = 2c Likewise, a -j- b — c = 2 (s — c).a — 6 -f c = 2 (s — 6). SOLUTION OF OBLIQUE TRIANGLES. 75 also, • 2 i a (s — b)(s — c)sin2 hA = * ^ J- bc (s —c)(s — i n S (S — C) cos- i 0= —^ —A #6 J also, These formulae may also be derived like those of § 56, bystarting with the equation, V _i_ c2 — a2 COS Yi = , 26cand adding both
Size: 1533px × 1629px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1890, booksubjecttrigonometry, bookyea
