TRIGONOMETRY. 21 
(supplemental triangle). Hence it follows that the sum of the angles of a 
spherical triangle must be greater than two right angles, and less than six. 
A spherical triangle is called right angled, when at least one of its sides is 
a right angle. If the triangle abc (fig. 113) be right angled at c, and we 
produce the sides ab and cb to d and e, so that ad = ce = 90°, and unite 
d and e by the are of a great circle, then bde is called the complemental 
triangle of abc, and de+the angle bac=90°; as also bed+the side 
We. 90°. 
The sines of the sides of a spherical triangle are to each other as the 
sines of the opposite angles. Let abc (fig. 114) be a spherical triangle, 
whose sphere has its centre in 0, and unity for radius. If now from c, on 
the plane aob, we let fall the perpendicular cd; from d on ae, bo, the 
perpendiculars de, df, and draw ce, cf; it would be easy to show that the 
triangles ceo, cfo are right angles, and consequently that ce = sin. cod, 
== Sin. areca; (cf = "sin. cab = sin. are cb. 
One of the most important formule in spherical trigonometry is that 
which expresses the cosine of an angle of a triangle, in terms of the three 
sides. To obtain this formula we may employ jig. 115, where abc is 
a spherical triangle, o the centre of the sphere, cd and ce tangents to the 
sides ca and cb, meeting the radii oa and ob ind ande. Drawing de, then 
according to a proposition of plane trigonometry, ve cd +. ce —2cd. 
ce. cos. dee; and also=— od’ + oe —2od. oe. cos. doe. But (indicating the 
radius by r) cd=r. tang. ac; ce=—r, tang. bc; angle dce = angle acb— c; 
r 
od= asa oa ed doe = ab. Substituting these values, we 
cos. ab — cos. ac. cos. be gil , 
ae : . If we indicate, as is customary, 
sin. ac. sin. be 
the angles by the capital letters A, B, C, and the sides corresponding to 
these letters by a, b, c, respectively, the preceding formula becomes 
cos. Cc — cos. a, cos. 6 a) 
cos. C = +. —__.._,  .. _If, however, we indicate the ‘sides ang 
sin. a@ —sin. 6 : 
angles by small letters, so that the side a’ answers to the angle a, &c., then 
cos. c' — cos. @’, cos. 5’ 
have cos. acb = 
Cus ¢ ==" These formule are not suited to calcu- 
sin. a’ sin. b’ 
lations of angles by means of logarithms. 
Two simple rules may be adduced, of universal application in calculating 
right angled spherical triangles. If, for instance, we write down the sides 
and angles of one of these in their natural order of sequence, omitting the 
right angle altogether, and taking for each side about the right angle, 90— 
that side, we shall have, 1, the cosine of any part—the product of the 
cotangents of the including parts, and 2, the cosine of any part = the 
product of the sines of the second and third parts following. Thus, if ¢ be 
the right angle, and we take b’ for 90 — }, and a’ for 90 —a, we shall have 
as the order of succession, a’, B, c, A, b’, a’, B, c; then, for example, 
cos. a’ = cot. B, cot. b’; and cos. A = sin. B, sin. a’, &c. The solutions 
thus obtained may be ambiguous when a part is given by its sine, since any 
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