A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Oblique-angled Triangles. 65
EXERCISES.-XX.
1. Prove that
cos (a + b) cos (a - b) tan C = sin a cos B + cos sin b. (278)
2-7. Solve a right-angled triangle, being given-1~. c, a+ b; 2~. c, a-b;
3~. a, b+c; 4~. a, b-c; 5~. c, A-B; 6~. c, p.
8. If 1 t cos a + cos b + cos c = 0, prove that each median is the supplement of the corresponding side.
9. In the same case, prove that the spherical excess is two right angles.
10. In the same case, prove that the arcs joining the middle points of
two sides are each = 90~.
11-17. If a + b + c = ir, proveA
1~. cos a = tan 1 B tan ~ C. 2. sin2 = cot b cot c.
3. cos2 A 4. tan2 a A cos b cos c
2 sin b sin c 2 cos a
A B C2
5~. sin2- + sin2- + sin2 = 1. 6. cos.A + cos B + cos = 1.
2 2 2
7~. cosec (A - E) + cosec (B - E) + cosec (C - E) = cosec E.
18. ABC is a spherical triangle right-angled at C; if with A, B as poles
great circles LHFKL, DEFG be describcd, meeting the sides CA, CB, AB
of the triangle in the pairs of points E, H; K, G; L, D, respectively;
prove that the five triangles ABC, ADE, HEF, FGK, KLB have all the
same circular parts.-(NAPIER.)
19-22.-Deduce from the analogies of DELAMBRE or NAPIER the following convergent series:a-b c A B A B
1~. = - cot - tan - sin c + 1 cot2 -tan2 sin 2c - &c.
2 2 2 2 2 2 2
(BRUNNOW). (279)
2~. - + cot- tan sin (a - b) + cot2 - tan2- sin 2(a- b)+&c.
2 \2/ c 2 2
(Ibid.) (280)
a+b c A B A B
30. = tan - tan sin c + tan2 tan2 - sin 2e + &c.
2 2 2 2 2
(Ibid.) (281)
c +b. B+b
4~. -tan tan sin (a+ b) +tan tan2 sin 2 ( + b)-c.
2 22 2 2
(Ibid.) (282)
F