
Applications
of Trigonometry 

Oblique
or Scalene Triangle 
Mollweide's formulas, the tangent
law (or the tangent rule) and halfangle formulas 
The tangent law or the tangent rule 
Halfangle 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 halfangle 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 


Halfangle 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 semiperimeter 

then 
A
= r
· s 
the
area of a triangle in terms of the 

semiperimeter and
inradius. 



The parts of a triangle denoted as in the diagram relates as
follows, 
x_{A}
+ x_{B}
= c,
x_{A}
+ x_{C}
= b,
x_{B} + x_{C}
= a
then, 2x_{A}
= 
(x_{B} + x_{C}) + b + c =

a + b + c, 
2x_{B}
= a 
(x_{A} + x_{C}) + c =
a 
b + c
and 2x_{C}
= a + b 
(x_{A}
+ x_{B}) + b + c = a + b

c, 
again
using a
+ b + c = 2s
it follows that x_{A}
= s 
a,
x_{B}
= s 
b
and x_{C}
= s 
c. 
Then, from the right triangles in the diagram, 

Equating obtained formulas with the halfangle 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 =
Ö3
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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