Plane trigonometry, by S.L. Loney.

SIDES AND ANGLES OF A TRIANGLE. 177
Let 2s stand for a + b + c, so that s is equal to half the
sum of the sides of the triangle, i.e. s is equal to the semiperimeter of the triangle.
We then have
a + b - c = a +b+c- 2c = 2s- 2c = 2 (s-c),
and a-b+c=-a+b+c-2b=s2s-2b=2(s-b).
The relation (1) therefore becomes
sin 2 (s - c) x 2 (s - b) (s-b) (s-c)
2 s -n _ = 2
-2 2be be
sin (sb( c.........(2)
2 bc
Similarly,
B (s - c)(s - a) C /(s - a) (s - b)
sin s = ca
2Zy CC~a 2 ab
166. To find the cosines of half the angles in terms of
the sides.
By Art. 109, we have
A
cos A = 2 cos2 A- 1.
2
A b2 + c2 - a2
Hence 2 cos2 1 +cosA = 1 +
2 2be
2bc + b2 + c2 - a2 (b + c)2 -a2
2be 2bc
[(b + c) + a] [(b + c) - a] (a + b + c)(b + c - a)
2bc 2bc
Now b +c-a=a+ b + c- 2a = 2s-2a=2 (s- a),

L. T.

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