There are a lot of formulas and techniques to find the area of a triangle. We can use many different formulas to calculate area of a triangle according to the given conditions. Here we shall derive some of the main formulas used to calculate area of a triangle.
Formulas for Area of a Triangle:
The area of a triangle is denoted by the symbol delta ( )
We shall appeal to the formula:
And the half
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Half Angle formulas?:
The Half angle formulas are stated below:
If ABC is a triangle , A , B and C are the three angles of the triangle and a , b , c are the sides opposite to the corresponding angles and
“s” is the semi perimeter or , , Then:
Proof of Half angle formula:
First of all let’s prove the half angle formula for
Using the cosine law:
Now usin
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Projection Law:
Projection law states that in any triangle:
Where , A , B , C are the three angled of the triangle and a , b , c are the corresponding opposite side of the angles.
Projection law or the formula of projection law express the algebraic sum of the projection of any two side in term of the third side.
Proof of Projection law:
To prove the projection law we shall take the
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Sine Law:
Sine law states that in any triangle ABC:
And also with some mathematics we can also prove the following :
Which is also closely related to the sine law.
Where , a , b and c are sides of a triangle , A , B and C are angles opposite to sides a , b, and c correspondingly and R is the circum-radius of the triangle as shown in following figure:
sine law
Proof of Sine
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Trigonometric Transformation Formulas:
The set of formulas which are useful in transforming sums and difference of trigonometric functions into their products and vice versa.
These sets of formulas are derived directly from Trigonometric Addition and Subtraction formulas.
Here we will derive the transformation formulas using following four formulas which are Trigonometric addition and subtraction
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Trigonometric multiple and sub-multiple angle formulas:
Prerequisite: Please consider studying following topics before you study this article for better grasp and understanding:
Trigonometric addition and subtraction formulas
Trigonometric Functions
Pythagorian Identities
In this tutorial we shall derive formula for trigonometric functions of multiple and sub-multiple angle , For exampl
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Trigonometric Addition ( Sum ) and Subtraction ( Difference ) formula:
The formulae
which are popularly
known as addition ( sum )
and subtraction( difference )
formulae are as follows:
Sine of sum of angles:
Cosine of sum of angles:
Sine of difference of angles:
Cosine of difference of angles:
And similarly the sum and difference of angle formula of Tangent are:
an
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Basic Distance Formula:
The basic distance formula states that:
The distance “d” between two points A(x1,y1) and B(x2,y2) can be calculated as:
Using this Distance Formula of coordinate geometry we can establish fundamental trigonometric formulae for general angles in a very elegant way. So we shall now prove or derive this formula:
Derivation or Proof of Distance Formul
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Right Angled Triangle:
In a
right angled triangle,
ABC with sides
BC = a
CA = b and
AB = c ;
right angled triangle
We know that:
or ,
And or ,
And or ,
It is obvious that SIN A = COS B , COS A = SIN B because:
A+B = 90 , So A is the complement angle of B , this may be stated as:
Sine of the angle A = Sine of the complement of B
So , Sine of the angle A = CoSine of
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Trigonometric functions of negative angles:
let and be any two angles equal in magnitude but opposite in sign.
If we place each of them in the standard position it can be observed that the two angles are symmetrically placed on either side of x-axis.
Suppose we construct a circle of radius “r” with centre “o” , it will cut the terminal arms of angles and
as shown
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