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 angle formula:

etc.

Where “s” is the semi circumference of the triangle or,

We shall now derive different formula for the area of triangle:

First formula for the area of a triangle:

Second formula for the area of a triangle:

Now , as 2s = (a+b+c)

Third formula for the area of a triangle:

Now using sine law:

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 using the trigonometric sub-multiple angle formula:

Now , let us prove the half angle formula for

Using the cosine law:

Lastly , Dividing by we get :

Similarly we can also prove the half angle formula of angle B and C.

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 help of sine law which states that:

Thus ,  using the above formula for sine law we can easily deduce the formula for projection law as:

Now using the trigonometric addition formulas we can replace by

And again in a triangle A+B+C = 180, so:

Now we can re write the original formula as:

And we can also deduce the second and third formula for projection law similarly.

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 Law:

Let us consider a Triangle ABC  , placed in the standard position with the vertex A at the origin and side AB along the positive x-axis as in the figure below:

Now,

* The co-ordinates of A are ( 0 , 0 )

* The co-ordinates of B are ( c , 0 )

* The co-ordinates of C are ( b cosA , b sinA )

Now ,

Area of the triangle ABC is:

Similarly if we place the other angles B and C in the standard position or origin the we also get the following by similar method:

Area of triangle =

By combining these three formulas we get:

Hence:

Now , we shall prove the sine law with another approach to find the sine relation or connection between the radius of circum-circle with the sine of angles and their opposite sides.

Or now we shall prove:

To prove this we denote the circum-centre of the triangle ABC by “O”.

There might be three cases in which angle “A” is either acute (fig. a) , obtuse (fig. b) or right angled (fig. c).

In any case let us join the circum-centre of the triangle “O” to B and produce it in another side up to “D” as shown in figures blow:

sine law

In first two cases above:

angle BDC = 90 degrees ( Because angle made on circumference from diameter is always 90 degrees)

Thus:

Or ,

And in the third case it is obvious that : because in this case angle A is 90 degrees and sine 90 = 1.

Thus we can now conclude:

Cosine Law states that in any triangle following equations can be implied:

Where “a” is the side opposite to vertex A ,  ”b” is the side opposite to vertex B and “c” is the side oposite to vertex C , or  ”a” , “b” , “c” are the three sides of a triangle and “A” , “B” , “C” are three angles of the triangle.

Derivation of Cosine Law:

We can prove the above stated equations by following way:
To prove the first of these formulas , we place the triangle ABC in the standard position with the vertex A at origin and the side AB along the positive x-axis. Then, The co-ordinates of three vertices A , B , C are :

(0,0) , (c,o) and (bCosA, bSinA) Respectively. The positive sign is to be taken if vertex C is above the x-axis and negative sign if it

is below x-axis.
This two figures describes how the co-ordinates ot the vertices A , B and C are defined.

Now , using the distance formula , We have:

After simplifying we get

or,

Thus we get ,

Thus we proved or derived the first formula among three formulas of cosine law by simillar method we can also prove second and third one. To prove the second and third formula of the cosine law we need to place other Vertices “B” or “C” in the standard position or at the origin , then repeat the calculations.