Cosine Theorem

Hi guys! I have a formula in mathematics, but I know it is very popular.

Here the formula:     b² = a² + c² – 2ac cos B

We know this formula as Cosine Theorem, but how the formula came?

Here I show the proving:

Let’s see to the picture,

By Pytagoras Theorem, we got:

h² = b² – (c – y)²  …………..(i)    also we got,           h² = a² – y² ……………………..(ii)

Then we got,    cos B = y/b à y = b cos B …………………………………………….(iii)

Because (i) and (ii) are same, so we got:

h² = h² …………………………………………………..(same)

b² – (c – y)²  = a² – y² …………………………………..(substitute from h²)

b² – (c² – 2cy + y²) = a² – y² ……………………………(after simplify (c – y)² )

b² – c² + 2cy - y² = a² – y² ………………………………(open the brackets)

b² = a² + c² – 2cy ………………………………………..(after add c² and y², reduce 2cy in all parts)

b² = a² + c² – 2ac cos B ....................................................(substitute y from (iii) )

You may prove for a and c as subject of the formula..

Thank you, and good luck! :)