(cos A cos B+ cos C) sin C [cos A cos B — cos (A + B)] sin C

3. a2 = b2 + c2, a2 = b2 + c2 - 2bc, a2 = b2 + c2+2bc.

(ii.) a + b = (a - b) (2 +√3), an isosceles triangle with the angles 30°,

EXERCISE XIII.

12. A = B=33° 33′ 27 ', C=112° 53′6′′. 17. 4° 23′ W. of N., or W. of S.

128. 171 miles; 32° 44′ W.

129. N. 36° 52′ W.; 36° 8′ W.

130. 173 miles; 51° 16' S.; 34° 13′ E.

131. S. 50° 58′ E.; 47° 15′ N.; 20° 49′ W.

132. N. 53° 20′ E., 16° 7′ W.; or N. 53° 20′ W., 25° 53′ W

133. N. 47° 42.5′ E., 19° 27′ N., 121° 51′ E.; or N. 47° 42.5′ W., 19° 27′

N., 116° 9′ E.; or S. 47° 42.5′ E., 14° 33′ N., 121° 48′ E.; or S

47° 42.5′ W., 14° 33′ N., 116° 12′ E.

137. N. 73 E., 45 miles; 42° 15′ N., 69° 5′ W.

138. N. 72° W., 287 miles; 33° S., 13° 2′ E.

90°,

3. (i.) Either a or b must be equal to 90°. (ii.) If a = 90°, then A = 90°, and B-b; if b = 90°, then B=90°, and A= =α. (iii.) c = A = 90°, B = b. (iv.) c= · 90°, A = 90°, B = 90°, C'= 90°.

EXERCISE XXI.

2. I. The cosine of the middle part = the product of the cotangents of the adjacent parts.

=

II. The cosine of the middle part the product of the sines of the opposite parts.

26. a = 90°, b = 45°, B = 45°.

27. a = 60°, b = 90°, B = 90°.

28. The triangle is impossible; why?

b

==

133° 39' 48".

29. b 130° 41′ 42′′, c = 71° 27′ 43", A 112° 57' 2".

=

30. a = 26° 3′ 51′′, A = 35°, B 65° 46' 7''.

31. Impossible; why?

EXERCISE XXIII.

=

1. cos A cot a tan b, sin B

= csc a sin b, cos h = cos a secb.

2. sin Aseca, or cos A = cos a sec2 2a, or tan & A sec a cosa.

4. Tetrahedron, 70° 31′ 43′′; octahedron, 109° 28' 18'; icosahedron, 138° 11′ 36′′; cube, 90°; dodecahedron, 116° 33′ 44′′.

5. cot A =√cos a.