The Cosine Law





Statement of cosine law:

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, \pm 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.

 

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