The Trigonometrical Formula Associated With Triangles.
There are a number of equations associated with triangles. Of these, the best known are the Sine and Cos formulae.
The Sine Formula.
Consider the Triangle with its Circumcircle.
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices , , and is denoted .
Acute Triangle
Draw the diameter through
Angle degrees and
From the diagram it can be seen that
Therefore by symmetry:
Obtuse Triangle
Triangles can be classified according to the relative lengths of their sides:
Equilateral triangle all sides have the same length
Isosceles triangle, two sides are equal in length
Scalene triangle, all sides are unequal
If is obtuse, angle
Acute Triangle
is an acute-angled triangle of height
Then
therefore
Using Pythagoras:
and
therefore
or
NOTE This equation can be re-written in terms of either angle or
Obtuse Triangle
Triangles can also be classified according to their internal angles :
A right triangle, has one of its interior angles measuring
A triangle that has one angle that measures more than is an obtuse triangle
A triangle that has all interior angles measuring less than is an acute triangle
If is obtuse
and
or
Thus
Note It should be noted that the same equation can be applied in both cases.
Two Additional Formulae For The Solution Of Triangles
The cos and sine formula together are sufficient to solve any triangle but the cos formula can be unwieldy in use and is sometimes replaced by the following:
Formula1
Using the sine formula
as and are complements
This is the quickest way of solving a triangle given two sides and the included angle.
Example:
Example - Example 1
Problem
If ; and is , find the other side and the other angles.
Workings
Thus
But
From which
From the sine formula
Solution
Formula2
Half Angle Formula
The cos formula can be used to find the ratios of the half angles in terms of the sides of the triangle and these are often used for the solution of triangles, being easier to handle than the cos formula when all three sides are given.
but
Let
or
where is the semi perimeter of the triangle
Similarly
By division
Area Of A Triangle
Let be a triangle
The area of a triangle is a half base times height.
The Median And Centre Of Gravity ( By Apollonius )
Also
The Orthocentre
Let be a regular triangle
and let be the orthocentre of the triangle
Using the sine formula for the triangle
But and
Similarly
But
Therefore
The Angle Bisector
Let and such as is the bisector of the
angle
As bisects the angle internally
But
And since
Applying the sine formula to
Therefore
The Angle bisector
The Pedal Triangle
The Pedal Triangle of ABC is the triangle formed by joining the feet of the altitudes of the triangle ABC.
If All The Angles Are Acute
Since is cyclic
And Since is cyclic
By addition
is the incentre of the Pedal Triangle and the angles are given by:
Note The sides of the Pedal Triangle are ; ; and
If A Is Obtuse
Since is cyclic,
Since is cyclic,
And therefore
Similarly
By subtraction
is the Incentre of the Triangle and the angles of the Pedal Triangle are:
Using the sine formula for triangle
:
Hence
Thus
or
The sides of the Pedal Triangle are ; ; Note It is worth knowing that in the case of either an acute or an obtuse angle triangle, the four points and are the three ex-centre and incentre of the Pedal Triangle.
The Circumcircle
The radius of the circum-circle can be obtained from:
from which it is possible to write:
The Incircle
The Area of the triangle is the sum of the areas of te triangles ; ; .
Similar equations can be written for triangles and
Therefore the area of triangle is given by:
where is the semi perimeter
If are the points of contact between the triangle and circle, then
; ; and the semi circumference of the triangle( ) is given by:
but
A similar relationship exists for ; etc.
For the triangle
Applying the sine formula to triangle
Thus
The Ex-circles.
The diagram shows the ex-circle opposite to . There are of course two more circles opposite and . There are similar equations for them .
Let , , , be the points of contact between the lines which make up the sides of the triangle and the circle.
Equating the areas of triangles , , , . we get:
This can be re-written in terms of the radius of the ex-circle
Thus the radius of the ex-circle, is given by the equation:
By inspection ; ; .
From triangle
Using the sine formula in triangle
The Triangle Formed By The Three Ex-centres
It is clearly possible to draw a triangle based upon the three ex-centres.
Since are the external bisectors of the angle , the line is a straight line as is also the line ( the Internal bisector)
As the external and internal bisectors of an angle are perpendicular, is perpendicular to
Therefore the triangle is the Pedal Triangle of and is the orthocentre.
Since
and
therefore
Thus
Therefore the Length of the side of the Triangle through is :
The nine point circle of will pass through the feet of its altitudes . The radius of the its nine point circle is therefor . But since the radius of a nine point circle is half that of the circum-circle, the radius of the circum-cirle of is
Hence
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