the third angle.
These theorems, which are given
in (749) and (758), can be obtained from equations
(702-704) by the assistance of the following lem-

This equation is the same as the proportion

sin. M cos. N: cos. M sin. N::x:y;

hence, by the theory of proportions,

sin. M cos. N + cos. M sin. N: sin. M cos. N

- cos. M sin. N:: x + y : x - y,

(739)

(740)

(741)

or, by (84) and (90),

sin. (M+N): sin. (M-N) :: x + y : x - y; which may be written in the form of an equation as in (737).

(742)

(746)

(747)

This equation is the same as the proportion

cos. M cos. N: sin. M sin. N : : y : x;

hence, by the theory of proportions,

cos. M cos. N - sin. M sin. N:cos. M cos. N + sin. M sin. N : :y - x: x + y,

or, by (104) and (116),

(748) cos. (M+N): cos. (M-N) : : y-x : y + x; which may be written as in (744).

60. Theorem. The sine of half the sum of two angles of a spherical triangle is to the sine of half their difference, as the tangent of half the side to (749) which they are both adjacent is to the tangent of half the difference of the other two sides; that is, in the spherical triangle ABC (figs. 4. and 5.),

(750)

B

sin. + (A + C): sin. + (A-C) Fig.4
:: tang. 16: tang. (a-c).

a

C

C

But we have by (228), accenting the letters so as not

to confound them with the angles of the triangle,

sin. + (A + C)

=

tang.b

sin. (A - C) tang. (a - c)'

which is the same as (750).

(757)

61. Theorem. The cosine of half the sum of two angles of a spherical triangle is to the cosine of half their difference, as the tangent of half the side to (758) which they are both adjacent is to the tangent of half the sum of the other two sides; that is, in the spherical triangle ABC (figs. 4. and 5.)

cos. (A + C): cos. (A-C): : tang.b

: tang. + (a + c).

(759) (760)

(761)

Demonstration. The product of (702) and (704),

is, by a simple reduction,

tang. A tang. + C

sin. (s - b)

sin. s

Hence, by (743) and (744),

cos. (A + C) sin. s

=

sin. (s-b)

cos. (A-C) sin. s + sin. (sb)

But (753) is, when inverted,

sin. A' sin. B

sin. A' + sin. B

If in this equation we make

(B'=s-b = (a - b + c);

tang. (A - B)

(762)

tang. (A + B)

§ A' = s = (a+b+c),

(763)

(765)

(766)

sin. s + sin. (s - b) tang. + (a + c)

This equation, substituted in (761), gives

which is the same as (759).

62. Scholium. In using (749) and (758), the signs (767) of the terms must be attended to by means of (496).

EXAMPLES.

1. Given in the spherical triangle ABC (figs. 4.

2. Given in the spherical triangle ABC (figs. 4.

and 5.)

A = 170°, C = 2°, b = 92° ;

to find a and c.

Ans. a = 103° 7',

11° 17'.

63. Theorem. The greater side of a spherical

triangle is opposite the greater angle.

(768)