EXAMPLES.

1. In the triangle ABC prove (1)

a+b:c= cos (A - B): sin C,

and (2)

ab: c = sin (A – B) : cos 1⁄2 C.

2. If AD bisects the angle A of the triangle ABC, prove

3. If AD' bisects the external vertical angle A, prove

99. To express the Sine, the Cosine, and the Tangent of Half an Angle of a Triangle in Terms of the Sides.

then

2 bc

a2 − (b − c ) 2

=

2 be

(a + b − c) ( a − b+c).

2 bc

(Art. 49)

a+b+c=2s;

a+b-c=2(sc), and a-b+c=2(s—b).

Since any angle of a triangle is <180°, the half angle is <90°; therefore the positive sign must be given to the radicals which occur in this article.

100. To express the Sine of an Angle in Terms of the

In any triangle ABC prove the following statements:

1. a (b cos C-c cos B) = b2 - c2.

2. (b+c) cos A+ (c+a) cos B+ (a + b) cos C = a+b+c.

7. a sin (BC) + b sin (CA) + c sin (AB) = 0.

9. tan Atan B (sb) ÷ (sc).

=

circumscribed circle of a triangle in terms of the sides of the triangle.

Let O be the centre of the circle A described about the triangle ABC, and R its radius.

Through O draw the diameter CD and join BD.

b

.. BC= 2 R sin A, or a = 2 R sin A.