209. Given a = - 68° 20′ 25′′, b = 52° 18′ 15′′, C=117° 12′ 20′′.

(a - b) = 8° 1′ 5′′, (a+b) = 60° 19′ 20′′, C=58° 36′ 10′′.

since sin C is greater than sin A we shall obtain two values for c both greater than a, and we shall not know which is the value to be taken.

We shall therefore determine c from formula (1) of Art. 54, which is free from ambiguity,

Here cos C is negative, and therefore tan will be negative, and greater than a right angle. The numerical value of cos C is the same as that of cos 62° 47′ 40′′.

Here cos is negative, and therefore cos c will be negative, and c will be greater than a right angle. The numerical value of cos is the same as that of cos (180° - 6), that is, of cos 30° 36′ 33′′; and the value of cos (a-0) is the same as that of cos (0-a), that is, of cos 81° 3′ 2′′.

Thus by taking only the nearest number of seconds in the tables the two methods give values of c which differ by 1"; if, however, we estimate fractions of a second both methods will agree in giving about 43 as the number of seconds.

210. Given a = 50° 45′ 20′′, b = 69° 12′ 40′′, A = 44° 22′ 10′′.

In this case there will be two solutions; see Art. 86. We will calculate C and c by Napier's analogies,