CHAPTER VI.

FORMULAS RELATING TO OBLIQUE TRIANGLES.

OF OBLIQUE TRIANGLES.

§ 71. The sides of any triangle are proportional to the sines of the opposite angles.

FIG. 41.

FIG. 42.

A A

C

BA

From B and C (Figs. 41 and 42) let fall the perpendiculars p' and p on b and c respectively. Then

In applying the above to Fig. 42, we must remember that sin B is equal to the sine of its supplement CBP.

§ 72. The sum of any two sides of a triangle is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.

From [67] we get, by the theory of proportions,

§ 73. The square of any side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of those sides into the cosine of their included angle.

Let w denote an angle of a parallelogram, m and n its sides, d the diagonal which divides w, and w, the supplement of w.

By constructing a proper figure it can be readily seen that d, m, and n form a triangle in which the angle included by m and n is w1.

.. by [69],

d2 = m2 + n2 — 2 m n cos w1 ;

but since w and w1 are supplements, we have, —

§ 74. Formula for the side of a triangle in terms of the cosines of the adjacent angles and the other two sides. From Figs. 43–45, it is evident that,

a=b cos C+c cos B,

b=ccos A+ a cos C,

[71]

ca cos B+b cos A.

From [71] all the relations between the six parts of a triangle can be deduced by algebraic transformations; but the processes are somewhat longer than those given in the

text.

§ 75. Formulas for sine and cosine of A, B, and C. From [69] we find

a2 — ( b 2 — 2 b c + c 2 ) _ (a−b+c) (a+b—c) ̧

put 2s=a+b+c,.·.a−b+c=2(s—b)

and a+b-c 2 (s—c)

* A, B, and C are less than 180°, and A, B, and C are less than 90°: we must then give the + sign to the radicals.

Since the sine of an angle and the sine of its supplement are the same (v. [8]), whenever all that is given concerning an angle is the value of its sine, the angle may have either of two supplementary values. The ambiguity thus arising in the use of [72] is, however, removed by the consideration, that, since A, B, and C, being angles of a triangle, are each less than 180°, A, B, and C are each less than 90°.

2

§ 76. Formulas for tan 1⁄2 A, tan 1⁄2 B, and tan 1⁄2 C, in terms of s, a, b, and c.