CHAP. IX.

On the Expressions for the Cosine and Sine of the Angle of a Spherical Triangle in Terms of the Sines and Cosines of the Sides.

We will begin this Chapter with establishing in the following Problem, the fundamental formula, from which all the methods and forms of solution will be deduced: it corresponds to the fundamental one of Plane Trigonometry, inserted in page 24; and the Student who understands, in principle, the use made of that latter formula, possesses, in fact, the clue to the subsequent demonstrations of Spherical Trigonometry.

PROP. XV. PROBLEM.

It is required to express the cosine of the angle of a spherical triangle in terms of the sines and cosines of the sides.

Let the triangle be mtn, let the sides be a, b, c, and the

opposite angles A, B, C; conceive O to be the centre of the sphere, and draw the two tangents t Q, t P, at the point t, to the arcs tm, tn; then ∠QtP = ∠A: by Prop. 6; and the angle at O is measured by mn, or a; and, by the definition of the secant,

OQ is sec. b
OP is sec. c.

The principle of the demonstration that follows, is, to obtain two values of QP, one from the triangle POQ, the other from PtQ, and then to compare them: now by Euclid, Prop. 13, Book II, or Prob. 1 of this Treatise :

in A POQ, PQ2 = sec.2 b + sec.2c-2 sec. b. sec. c cos. a [rad. 1] in & PtQ, PQ2 = tan.2b+tan.2 c - 2 tan. b.tan.c.cos. A.

Subtract the lower expression from the upper,

then, since sec. b - tan. b = (rad.) = 1, we have 0=1+1+ 2 tan. b.tan.c.cos. A 2 sec. b. sec. c. cos. a

and similarly, since the process for finding cos. B, cos. C, will be exactly the same, changing a for b, b for a, &c. the result must be similar, that is,

cos. c = sin. a. sin.b.cos. C + cos. a. cos. b

by substituting this value of cos. c in the expression [a], we

have

COS. a

cos.

cos.b.sin.b.sina.cos C - cos. a cos.2 b

sin. b. sin.c

But, cos. a-cos. a. cos2b=cos. a (1-cos.2 b)=cos. a. sin.2b;

.. cos. A + cos. B

(cos. a. sin. b + cos. b. sin. a) (1 - cos. C)

It is required to express the sine of the angle of a spherical triangle in terms dependent on its sides.

By the last Proposition, cos. A

COS. a cos. b.cos.c

sin. b.sin.c

.. 1 + cos. A

;

cos.a-(cos.b.cos.c-sin.b.sin.c)_cos.a - cos. (b+c)

by the form [2], p. 130; but by the form [8], page 33,

cos. a - cos. (b + c) = 2 sin. (a + b + c) sin. (b + c -a).

2

2

or, = sin sinex sin. (a+b+c). sin. (a+b+c - )

2

b c

2

a

cos. b. cos. c + sin. b sin. c

cos.

(b-c) - cos.

sin. b. sin. c

(by the form [4], p. 31.)

2

sin. b sin. c

x.sin. (a+b)sin. (a+c-b) by form [8], p. 33;

2

2

or, = sin sixsin. (a+b+c_b) sin. (a+b+c).

2

sin.b.sin.c

2

2

Hence, if we multiply together 1 + cos. A, and 1 cos. A,

the product of which is sin.2 A, and substitute S instead of

4

× sin. S. sin. (S - a) sin. (S - b) sin. (S-с),

sin.2 b.sin.2 c

and consequently,

sin. A =

2 sin.b.sin.c

√{ sin. S.sin. (S - a) sin. (S - b) sin. (S - c) } *.

Cor. 1. If we wish to compute sin. B, we must begin cos. b-cos.a.cos.c from cos. B = sin, a. sın. c

the steps of the former process: the result will be a fraction, the numerator of which is the numerator of the above fraction for the sin. A, and the denominator will be sin. a sin.c; call the common numerator N, then

or (if these equations be expressed, after the manner of expres

* The Student, for his own convenience, is desired to compare these expressions and the manner of deducing them, with the corresponding ones in Plane Trigonometry.

sing a proposition) The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles.

Right-angled spherical triangles may be considered as particular cases of oblique. The solutions of the latter, then, would necessarily include those of the former; and, accordingly, if we wished to generalise as much as we could generalise, it would not be requisite separately to consider them. Since, however, it is our object to render investigation as simple and as easy as it is possible to the Student, we shall not avail ourselves of this abridgment, and seek to be compendious, but proceed, in the ensuing Chapter, to treat distinctly of the cases of right-angled spherical triangles.