. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 99) O .•. sinA = — Vs(s — a)(s — b)(s — c).be Similarly, sin B = — Vs (s — a) (s — b) (s — c),ac 2 sin C = — Vs (s — a) (s — 6) (s — c).ab Cor. sin A = — V262c2+2c2a2+ 2a262- a4- 64- c4 26c and similar expressions for sin B, sin C. EXAMPLES. In any triangle ABC prove the following statements: 1. a (b cos C — c cos B) = b2 — c2. 2. (6 + c) cos A + (c + a) cos B + (a + b) cos C = a + b + a o sin A + 2 sin B __ sin Ca + 2b c A sin2 A — m sin2 B sin2 C 4. ~— = — . a2 — mb2 <
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 99) O .•. sinA = — Vs(s — a)(s — b)(s — c).be Similarly, sin B = — Vs (s — a) (s — b) (s — c),ac 2 sin C = — Vs (s — a) (s — 6) (s — c).ab Cor. sin A = — V262c2+2c2a2+ 2a262- a4- 64- c4 26c and similar expressions for sin B, sin C. EXAMPLES. In any triangle ABC prove the following statements: 1. a (b cos C — c cos B) = b2 — c2. 2. (6 + c) cos A + (c + a) cos B + (a + b) cos C = a + b + a o sin A + 2 sin B __ sin Ca + 2b c A sin2 A — m sin2 B sin2 C 4. ~— = — . a2 — mb2 <? 5. a cos A + b cos B — c cos C = 2c cos A cos B. 6 cos A cosB cosC « sin B sin C sin C sin A sin A sin B ~ 7. a sin (B - C) + b sin (C - A) + c sin (A - B) = 0. 8. tan£Atan£B = ^i. s 9. tan£A-^tan£B = (s-&)-5-($--c). AREA OF A TRIANGLE. 153 101. Expressions for the Areaof a Triangle. (1) Given tivo sides and theirincluded angle. Let S denote the area of the tri-angle ABC. Then by Geometry, 28 = ex CD. But in either figure, by Art. 94, CD = b sin A. ,\ S = ^ Similarly, S = ^ ac sin B, S = \ ab sin C. (2) Given one side and the angles. Since which is a : b = sin A : sin B, (Art. 95) , a sin Bsm A S = ^ ab sin C, gives a2 sin B sin C Similarly, S =S = 2 sin Ab2 sin A sin C c2 sin A sin B 2sinB 2sinC (3) Given the three sides. sin A = iL V«(* - a) (s -b)(s- c) (Art. 100)oc Substituting in we get S = \ be sin A, S = Vs(s — a)(s — b)(s — c). 154 PLANE TBIGONOMETBY. 102. Inscribed Circle. — To find the radius of the inscribedcircle of a triangle. q Let ABC be a triangle, 0 thecentre of the inscribed circle, andr its radius. Draw radii to thepoints of contact D, E, F; and joinOA, OB, OC. Then A c D S = area of ABC = A AOB + A BOC + A COA = ±rc + ±ra + ± rb
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