1. ## cosine rule

hi!

can i know how to prove that the cosine rule is true for ∏/2≤ x ≤∏ for
c^2 = a^2 + b^2 - 2ab cos x?

i want to show that it is true for 0 ≤ x ≤∏ and have proven that it is true for 0≤x≤∏/2 but i dont know how to prove the other part..

thanks!!!

2. Hello alexandrabel90
Originally Posted by alexandrabel90
hi!

can i know how to prove that the cosine rule is true for ∏/2≤ x ≤∏ for
c^2 = a^2 + b^2 - 2ab cos x?

i want to show that it is true for 0 ≤ x ≤∏ and have proven that it is true for 0≤x≤∏/2 but i dont know how to prove the other part..

thanks!!!
In the attached diagram:

$\displaystyle c^2 = [a + b\cos(\pi-x)]^2+[b\sin(\pi-x)]^2$

$\displaystyle =a^2+2ab\cos(\pi-x)+b^2\cos^2(\pi-x)+b^2\sin^2(\pi-x)$

Now $\displaystyle \cos(\pi-x) = -\cos x$ and $\displaystyle \sin(\pi-x) = \sin x$. So $\displaystyle \cos^2(\pi-x) = (-\cos x)^2 =\cos^2x$

$\displaystyle \Rightarrow c^2 = a^2 -2ab\cos x + b^2(\cos^2x + \sin^2x)$

$\displaystyle = a^2 +b^2 -2ab\cos x$, since $\displaystyle \cos^2x + \sin^2x = 1$