SPHERICAL TRIGONOMETRY.

HAVING demonstrated in the treatise on Spherical Geometry, several important properties of the circle of the sphere, and of spherical triangles, we shall now proceed to deduce various relations which exist between the several parts of a spherical triangle. These constitute what is called Spherical Trigonometry; and enable us, when a certain number of the parts are given, to determine the rest. The first formula which we shall establish, serves as a key to all the rest, and is to spherical trigonometry what the expression for the sine of the sum of two angles is to plane trigonometry.

CHAPTER I.

1. To express the cosine of an angle of a spherical triangle in terms of the sines and cosines of the sides.

Let A B C be a spherical triangle, O the

centre of the sphere.

Let the angles of the triangle be denoted by the large letters A, B, C, and the sides opposite to them by the corresponding small letters, a, b, c.

At the point A, draw A T a tangent to the zrc A B, and A t a tangent to the arc A C.

Then the spherical angle A is equal to the angle T At between the tangents (Spher. Geom. prop. IV.).

Join O B, and produce it to meet A T in T.
Join O C, and produce it to meet A t in t.
Join T, t;

B

T

t

tan.c+tan. b— 2 tan, c tan. b cos. A sec.c+-sec.2 b·

=1+tan. c + 1
sec. b cos. a.

2 sec. c sec. b cos a. + tan.2 b— 2 sec, c

2. To express the cosine of a side of a spherical triangle, in terms of the sines and cosines of the angles.

Let A, B, C, a, b, c, be the angles and sides of a spherical triangle; A', B, C, a', b, c, the corresponding quantities in the Polar triangle,

Then, by (a),

cos. ď

cos. A' =

-cos. b' cos. c' sin. b' sin. c'.

But (Spherical Geometry, prop. VI.), A′ = (180° — a), a′ = (180° —— A), b' — (180o — B), c′ — (180o — C),

3. To express the sine of an angle of a spherical triangle, in terms of the sines of the sides of the triangle.

By (a) we have,

Again, resuming the expression for cos. A,

cos. b cos. c + sin. b sin. c— cos. ɑ

sin. b sin. c

1- cos. A

=

(1.)

Finally, dividing the expressions (y. 3 by those y. 2), we obtain,

4. To express the sine of a side of a spherical triangle, in terms of the sines and cosines of the angles.

sin. c =

sin. A sin. C

2

sin. A sin. BV —cos. s′ cos. (s'—-Â) cos. (s′—B) cos.(s′—C)