CASE II. When the included angle A is obtuse. In Fig. 3, BD = AD+c. Squaring, and adding CD to both members, But, Also, BD2 + CD2 = AD2 + CD2 + c2 + 2 c × AD. BD+CD a2, and AD2 + CD2 = b2. = 117. To express the cosines of the angles of a triangle in terms of the sides of the triangle. 118. To express the sines, cosines, and tangents of the half-angles of a triangle in terms of the sides of the triangle. Denoting a+b+c by 2s, so that s is the half-sum of the sides of the triangle, we have and ab+c=(a+b+c)-2b=2s-2b-2 (s—b), a+b-c=(a+b+c)-2c=2s-2c=2(s-c). Again, adding both members of (55) to unity, we have b+ca=(b+c+a)-2a=2(s-a). cos2 A = and Hence, or, In like manner, and 48 (s- a) 4 bc bc са cos C= ab (63) Note. Since each angle of a triangle is less than 180°, its half is less than 90°; hence the positive sign must be taken before the radical in each of the formulæ of Art. 118. |