Cor. The areas of the colunar triangles are (2A - E) (2 B — E) π2, 180° 220. Problem. given the three sides. 180° (2 C — E) πr2. 180° To find the area of a triangle, having Here the object is to express E in terms of the sides. I. Cagnoli's Theorem. sin Esin (A+B+C − π) =sin(A+B) sin C-cos (A+B) cos C sin C cos C [cos (a - b) — cos (a+b)] . (Art. 198) cos c = = √tan s tan 1 (s—a) tan 1⁄2 (s—b) tan (s—c) (Art. 195) (3) 221. Problem. To find the area of a triangle, having given two sides and the included angle. cos Ecos [(A + B) − († π − † C)] = cos(A + B) sin + C + sin (A+B) cos C = cos(a+b) sin2C+cos(a-b) cos2C (Art. 198) = [cosa cos b+sina sin b cos C] secc = (1) Dividing (1) of Art. 220 by this equation, and reducing, we have 2. find 3. find 4. find 75°33'; Given A 84°20'19", B = 27° 22' 40", C= 75° 33'; E=7°15' 59". Given a 46° 24', b = 67° 14', c = 81°12'; = Given a 108° 14', b = 75° 29', c = 56° 37'; 1+ cos a + cos b + cos c 5. Prove cos E = 4 cosa cosb cos c cos2a + cos2b+cos2 c-1. 2 cosa cosb cos c sinssin (s-a) sin (s—b) sin(s—c). cosa cosb cosc. cos scos (s-a) cos 1⁄2 (s—b) cos (s—c). cosa cosb cos c cota cot b+cos C sin C cot bcotc+cos A sin A cotic cota + cos B 7. 66 cos E= 8. "cot E = EXAMPLES. Prove the following: 1. sin(sa) + sin (s — b) + sin(s — c) — sins = 4 sina sinb sinc. 2. sins + sin(sb) + sin(sc) sin (sa) 3. sin(sb) sin (s — c) + sin(s — c) sin (s — a) +sin(sa) sin (sb) + sin s sin (sa) sins sin (sb) + sin s sin (sc) = sin b sinc+ sinc sina + sin a sin b. 4. sin(sb) sin (s—c) + sin(sc) sin(sa) = sin (sa) sin (s — b) + sin s sin (s—a) sins sin (sb) sins sin(s—c) sinb sinc+ sin c sin a sin a sin b. 5. sin's + sin(s — a) + sin2 (s — b) + sin2 (s — c) = 2(1― cos a cos b cos c). 6. sin's + sin(s — a) — sin2 (s—b) — sin2(sc) = 2 cos a sin b sin c. 7. cos2s+cos2 (s − a) + cos2 (s — b) + cos2 (s — c) 8. cos2s + cos2(s — a) — cos2 (s — b) — cos2 (s — c) 17. cot2r12+cot2r2+ cot2r; +cot2r = 2(1–cos a cos b cos () n2 19. cot r2 cot r2 + cot r2 cot r1 + cot r1 cot r2 n2 +cot r(cot r+cot r2+cot rs) sin b sin c + sin e sin a + sin a sin b n2 20. tan r2 tan r2+ tan r tan r1 + tan r1 tan r2 25. tan R1+tan R2+ tan R tan R = 2 cot r. 2 26. tan R tan R1 + tan R2+ tan R2 = 2 cot 11. 3 27. tan R+ tan R1 tan R2+tan R = 2 cot 12. 28. tan R+tan R1 + tan R2 tan R3 = 2 cot îo. 33. tan R+cot r = tan R1 + cot r1 = etc., = 1⁄2 (cot r + cot r1 + cot r2+ cot r3). = 34. tan2 R+tan2 R1 + tan2 R2 + tan2 R3 2(1+cos A cos B cos C) N2 |