Wednesday, September 26, 2018

sequences and series - The sides of a triangle are in Arithmetic progression




If the sides of a triangle are in Arithmetic progression and the greatest and smallest angles are $X$ and $Y$, then show that



$$4(1- \cos X)(1-\cos Y) = \cos X + \cos Y$$



I tried using sine rule but can't solve it.


Answer



Let the sides be $a-d,a,a+d$ (with $a>d)$ be the three sides of the triangle, so $X$ corresponds to the side with length $a-d$ and $Y$ that to with length $a+d$. Using cosine formula
\begin{align*}
\cos X & = \frac{(a+d)^2+a^2-(a-d)^2)}{2a(a+d)}=\frac{a+4d}{2(a+d)}\\
\cos Y & = \frac{(a-d)^2+a^2-(a+d)^2)}{2a(a-d)}=\frac{a-4d}{2(a-d)}\\

\end{align*}
Then
$$\cos X +\cos Y=\frac{a^2-4d^2}{a^2-d^2}=4 \frac{(a-2d)}{2(a+d)}\frac{(a+2d)}{2(a-d)}=4(1-\cos X)(1-\cos Y).$$


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