Solution of Triangles

Solution of Triangles

In a triangle ABC, the angles are denoted by capital letters A, B, and C and the lengths of the sides opposite these angles are denoted by a, b, c respectively.

Semi-perimeter of the triangle is written as and its area by S or .



Let R be the radius of the circumcircle of triangle ABC.


Basic Formulae

(i) Sine rule:

.

(ii) Cosine rule:



(iii) Projection rule:

a = b cosC + c cosB,

b = c cosA + a cosC,

c = a cosB + b cosA.

(iv) Napier's analogy:




(v) Trigonometric ratios of half - angles:







Since, every angle of a triangle is less than 180⁰, half of each angle will be less than 90⁰, and thus, all the trigonometric ratios of half the angles are positive.

(vi) Area of a triangle:




Solution of Triangles:

The three sides a, b, c and the three angles A, B, C are called the elements of the triangle ABC. When any three of these six elements (except all the three angles)

of a triangle are given, the triangle is known completely; that is the other three elements can be expressed in terms of the given elements and can be evaluated. This

process is called the solution of triangles.

(i) If the three sides a, b, c are given, angle A is obtained from or B and C can be obtained in the similar way.

(ii) If two sides b and c and the included angle A are given, then gives Also so that B and C can be evaluated. The third side is given by a =

or a² = b² + c² - 2bc cosA.

(iii) If two sides b and c and the angle B (opposite to side b) are given, then , A = 180° - (B + C) and give the remaining elements. If b < c sinB, there

is no triangle possible (fig1). If b = c sinB and B is an acute angle, then only one triangle is possible (fig 2). If c sinB < b < c and B is an acute angle, then there are

two value of angle C (fig 3). If c < b and B is an acute angle, then there is only one triangle (fig 4).



This is, sometimes, called an ambiguous case.

If one side a and angles B and C are given, then A = 180° - (B + C), and .

If the three angles A, B, C are given, we can only find the ratios of the sides a, b, c by using the sine rule (since there are infinite number of similar triangles).


Circles Connected with Triangle

(i) Circum-circle:

The circle passing through the vertices of the triangle ABC is called the circum-circle. Its radius R is called the circum-radius. In the triangle ABC,







(ii) In-Circle:

The circle touching the three sides of the triangle internally is called the inscribed or the in-circle of the triangle. Its radius r is called the in-radius of the circle. In

the triangle ABC,






Remark:
From r = 4R sinsinsin, we find that r 4R.

2r R. Here equality holds for the equilateral triangle.


(iii) Escribed circles:

The circle touching BC and the two sides AB and AC produced of ABC externally is called the escribed circle opposite A. Its radius is denoted by r₁.

Similarly r₂, and r₃ denote the radii of the escribed circles opposite angles B and C respectively.

r₁, r₂, r₃ are called the ex-radii of ABC. Here





,



,

r₁ + r₂ + r₃ = 4R + r,

.