Plane trigonometry, by S.L. Loney.

SIDES AND ANGLES OF A TRIANGLE.

185

EXAMPLES. XXVII.
In any triangle ABC, prove that. B-C b-c A
1. --— = COS2 a 2
2. a(cosB+cosC)=2(b+c)sin2-.
2
3. a (cos G - cos B) = 2 ( - c) cos2.
a+b A+B A-B
4.- =tan 2cot 2
5. (b+c-a) (ot +cot\ 2a cot.
6. a2 b + 2c2 =2 (be cosA+cacos B + ab cos C).
7. (a2 - b2 +2) tan B = (a2 + b2- c2) tan C.
C C
8. c2=(a b)2cos2 +(a +b)2 sin2.
2 2
9. a sin (B- C)+bsin (C-) + c sin (A -B) = 0.
a sin (B - C) b sin (C - A) c sin (A- B)
10. b2- c2 2-a2 a2 b2
A.B- C. B C-A C A -B
11. a sin - sin 2 +bsin- sin l + c sin - sin =0.
2 2 2 2 2 2
12. a2 (co2 B - c2 C) + b (cos2 C- cos22 A) + c2 (cos2 A - cos2 B)=0.
b2 _- 2 Ac2_ a2 a2 _ b2
13. - sin2A+ --- sin2B+ ---sin2C=-0.
a2 b2 C
A B C
cot -+cot + cotC
(a + b + c) 2 2 2
a2 + b2 + c2 - cot A + cot B + cot C'
15. a3 cos (B - C) + b cos (C - A) + c3 cos (A - B) = 3 abc.
16. In a triangle whose sides are 3, 4, and /38 feet respectively,
prove that the largest angle is greater than 120~.