§ 77. Formulas for the area of a triangle, in terms of two sides and the included angle; in terms of one side and the adjacent angles; and in terms of s and the three sides.

... denoting the area by K we have,

Substituting these values in.[75], and noting that, since A 180° — (B+C), sin A = sin (B+C), we

get, —

=

By [26],

sin A 2 sin A cos A.

... substituting in [75] we get,

Kbc 2 sin A cos A.

§ 78. Formula for the area of a triangle, in terms of s and r (the radius of the inscribed circle).

ABC

BOCCOA + AOB = ar+1br+1/cr.

.. K=(a+b+c) r=sr

[78]

$ 79. Formula for the radius of the inscribed circle, in terms of s, a, b, and c; and in terms of s, A, B, and C.

From [79] and [74] we get,

r = s tan A tan 1 B tan 1 C.

[80]

§ 80. Formulas for tan A, tan B, and tan C, in terms of r, s, a, b, and c.

Dividing both sides of [79] by sa, we get,

§ 81. Formulas for the perpendiculars Pa, Pr, and Per from the angles upon the sides a, b, and c respectively.

§ 82. Formulas for radius of circumscribed circle (R), in terms of a side and the opposite angle; and in terms of K, a, b, and c.

In Fig. 48 draw the radius OD perpendicular to BC; therefore by geometry,

NOTE. In each set of formulas numbered 67, 69, 71, 72, 73, 74, and 82, the first is perfectly general: from it, therefore, the other two can be obtained by direct substitution. The work of substitution is made easier by the following rule: Given the first in each set, the others are obtained by advancing the letters; that is, by substituting for each letter of the equation the next letter in the fixed order indicated by the figure in the margin. In this manner, the second of [74], for example, can be obtained from the first, and then

§ 83. Solution of oblique triangles.

"We may remark here, that when an angle of a triangle is determined from its cosine, versed sine, tangent, cotangent, or secant, no uncertainty can exist about the angle, because only one angle exists less than 180° for which any of these functions has an assigned value. But when an angle of a triangle is determined from its sine or cosecant, uncertainty may exist, since there are two angles less than 180° which have a given sine or a given cosecant."Todhunter.

There is, however, no uncertainty in the case of right triangles, because each angle, except the right angle, is acute

According to the methods of solution which we shall adopt, there is only one case of oblique triangles where there can be any uncertainty in the angle: this case will be fully discussed hereafter (v. Case II. § 85).

§ 84. CASE I.

Given two angles and a side,-A, B, and a. find C, b, and c.

A+B+C 180°, ... C=180°- (A+B).

=

the third from the second, and the first again from the third. In like manner, the second may be obtained from the third, the first from the second, and the third from the first, by moving the letters back.