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[SOLVED] Angles in a triangle

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anemone

MHB POTW Director
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Feb 14, 2012
3,812
All the angles in triangle $ABC$ are less than $120^{\circ}$. Prove that

$\dfrac{\cos A+\cos B+\cos C}{\sin A+\sin B+\sin C}>-\dfrac{\sqrt{3}}{3}$.
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,812
\begin{tikzpicture}

\coordinate[label=left:A] (A) at (0,0);
\coordinate[label=right:B] (B) at (6, 0);
\coordinate[label=above:C] (C) at (2.4,3);
\coordinate[label=above: $A_1$] ($A_1$) at (7,3);
\coordinate[label=below: $B_1$] ($B_1$) at (7.2,0);
\coordinate[label=below: $C_1$] ($C_1$) at (12,2);
\draw (A) -- (B)-- (C)-- (A);
\draw (7,3) -- (7.2,0)-- (12,2)-- (7,3);

\end{tikzpicture}
Consider the triangle $A_1B_1C_1$ where $\angle A_1=120^{\circ}-\angle A,\,\angle B_1=120^{\circ}-\angle B$ and $\angle C_1=120^{\circ}-\angle C$. The given condition guarantees the existence of such a triangle.

Applying the triangle inequality in triangle $A_1B_1C_1$ gives $B_1C_1+C_1A_1>A_1B_1$, i.e.

$\sin A_1+\sin B_1>\sin C_1$ by applying the law of sines to triangle $A_1B_1C_1$.

It follows that

$\sin (120^{\circ}-A)+\sin (120^{\circ}-B)>\sin (120^{\circ}-C)$ or

$\dfrac{\sqrt{3}}{2}(\cos A+\cos B+\cos C)+\dfrac{1}{2}(\sin A+\sin B+\sin C)>0$.

Taking into account that $a+b>c$ implies $\sin A+\sin B-\sin C>0$, the above inequality can be rewritten as

$\dfrac{\sqrt{3}}{2}\cdot \dfrac{\cos A+\cos B+\cos C}{\sin A+\sin B+\sin C}+\dfrac{1}{2}>0$, from which the conclusion follows.