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

156 Miscellaneous Exercises.
Miscellaneous Exercises.
1. Prove that in a right-angled spherical triangle
tan r = sin (s - c), tan r'= sin (s-b), tan r" = sin (s-a), tan r"' = sin s.
2. If the plane angles of a trihedral angle be respectively equal to the
angles of a square, a hexagon, and a decagon, prove that the sum of its
dihedral angles is five right angles.-(CATALAN.)
3. If Ai be a chordal angle of a spherical triangle ABC, prove
1 + cos a - cos b - cos c
cos -4A1 = --. (486)
4 sin b sin ( c
4. If a spherical quadrilateral be inscribed in a small circle of the sphere,
prove that the cosine of its third diagonal is equal to the product of the
cosines of the tangents drawn to the small circle from the extremities of the
third diagonal.
5. Prove that the volume of the pyramid whose summits are the angular
points of a spherical triangle and the centre of the sphere, if the radius be
equal to unity, is l3/tan r. tan ra. tan rb. tan rc.
6. Prove that the angles of intersection of IHAT'S circle with the sides
of a spherical triangle are (A - B), (B - C), (C - A), respectively.
7. If in a trihedral angle O-ABC we inscribe two spheres, which
touch each other, if R1, R2 be their radii, prove that
R2 4sin (s - a) sin(s - b) sin (s- c) 1 + sin (s - a) sin(s - b) sin(s -c) }
(Ri sin s sins }
(STEINER.) (487)
8. If any angle of a spherical triangle be equal to the corresponding
angle of its polar triangle, prove
sec2A + sec2B + sec2 C + 2sec A sec B sec C = 1. (488)
9. If ABC be a diametral triangle, of which the side c is the diameter,
sin2 C = sin2 a + sin2 b. (489)
10. Express sin s, sin (s - a), &c., in terms of the in-radii of a triangle
and its colunar triangles.
11. Express sinE, sin (A - E), in terms of the circumradii of a triangle
and its colunars.