Applications of Trigonometry
     Oblique or Scalene Triangle
      Mollweide's formulas, the tangent law (or the tangent rule) and half-angle formulas
         The tangent law or the tangent rule
         Half-angle formulas
      Area of a triangle in terms of the inscribed circle or incircle
         The radius of the inscribed circle
      Oblique or scalene triangle examples
Oblique or Scalene Triangle
Mollweide's formulas, the tangent law (or the tangent rule) and half-angle formulas
Applying the same procedure on the other propositions of the sine law we derive other Mollweide's formulas,
The tangent law or the tangent rule
Dividing corresponding pairs of Mollweide's formulas and applying following identities,
obtained are equations that represent the tangent law
Half-angle formulas
Equating the formula of the cosine law and known identities, that is, 
plugged into the above formula gives
dividing above expressions
Applying the same method on the angles, b and g, obtained are 
Area of a triangle in terms of the inscribed circle (or incircle) radius
The oblique triangle ABC in the figure below consists of three triangles, ABO, BCO and ACO with the same altitude r therefore, its area can be written as
where  is the semi-perimeter
then  A = r s     the area of a triangle in terms of the 
semi-perimeter and inradius.
The parts of a triangle denoted as in the diagram relates as follows,
xA + xB = c,    xA + xC = b,    xB + xC = a  then,   2xA = - (xB + xC) + b + c = - a + b + c,
2xB = a - (xA + xC) + c a - b + c  and   2xC = a + b - (xA + xB) + b + c = a + b - c,
again using    a + b + c = 2s   it follows that   xA = s - a,    xB = s - b  and   xC = s - c.
Then, from the right triangles in the diagram, 
Equating obtained formulas with the half-angle formulas, as for example
                                            or   the radius of the inscribed circle.  
Plugging given r into the formula for the area of a triangle  A = r s  yields
     Heron's formula.
Oblique or scalene triangle examples
Example:   Determine length of sides, angles and area of a triangle of which a + b = 17 cm, c = 15 cm and angle g =113.
Solution:  Using Mollweide's formula
Applying the sine law,
Area of the triangle from the formula  
Example:   Given is the sum of the sides of a triangle  a + b + c = 46 cm, the radius of the incircle (or inradius)  r = 3 cm and angle b =60. Find all sides and angles of the triangle.
Solution:  Using the formula
Using Mollweide's formula
Applying the sine law,
Example:   Determine the area of an isosceles triangle of which, the line segment that joints the midpoint of one of its equal sides by the midpoint of the base equals the half of the radius R of the circumcircle.
Solution:  In the similar triangles  ABC  and  CDE,    b/2 = R/2    =>   b = R.
 
Area of the triangle
Equating obtained formula with the known formula for the area of a triangle in terms of the radius of the 
circumcircle  
so, the area of the triangle 
Geometry and use of trigonometry contents - B
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