MHB Trigonometric Inequality Challenge

anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
For any triangle $ABC$, prove that

$\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$
 
Mathematics news on Phys.org
anemone said:
For any triangle $ABC$, prove that

$\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$

Soluion by David E. Narvaez, Panama:

From Jensen's inequality we have that

$\tan\dfrac{A}{2}+\tan\dfrac{B}{2}+\tan\dfrac{C}{2}\ge\sqrt{3}$ and $\sin\dfrac{A}{2}\sin\dfrac{B}{2}+\sin\dfrac{B}{2}\sin\dfrac{C}{2}+\sin\dfrac{C}{2}\sin\dfrac{A}{2} \ge\dfrac{3}{4}$ thus

$\displaystyle \dfrac{3}{2}\left( \sum_{cyc} \tan \dfrac{A}{2} \right)\left( \sum_{cyc} \sin \dfrac{B}{2} \sin \dfrac{C}{2} \right) \ge \dfrac{\sqrt{3}}{2}$

Let us assume, without loss of generality, that $A\ge B \ge C$. Then $\left( \tan \dfrac{A}{2} +\tan \dfrac{B}{2} \right) \ge \left( \tan \dfrac{A}{2} +\tan \dfrac{C}{2} \right) \ge \left( \tan \dfrac{B}{2} +\tan \dfrac{C}{2} \right)$ and

$\sin \dfrac{A}{2}\sin \dfrac{B}{2} \ge \sin \dfrac{C}{2} \sin \dfrac{A}{2} \ge \sin\dfrac{B}{2}\sin \dfrac{C}{2}$ and by Chebychev's inequality, we get

$\displaystyle \sum_{cyc} \left(\tan \dfrac{B}{2}+\tan \dfrac{C}{2} \right)\sin \dfrac{B}{2} \sin \dfrac{C}{2} \ge \dfrac{1}{3} \left(\sum_{cyc} \left(\tan \dfrac{B}{2}+\tan \dfrac{C}{2} \right) \right) \left( \sum_{cyc} \sin \dfrac{B}{2} \sin \dfrac{C}{2} \right) \ge \dfrac{\sqrt{3}}{2}$

but

$\begin{align*}

\left(\tan \dfrac{B}{2}+\tan \dfrac{C}{2} \right)\sin \dfrac{B}{2} \sin \dfrac{C}{2}&=\left(\dfrac{\sin \dfrac{B}{2} \cos \dfrac{C}{2}+\sin \dfrac{C}{2} \cos \dfrac{B}{2}}{\cos \dfrac{B}{2}\cos \dfrac{C}{2}} \right)\sin \dfrac{B}{2} \sin \dfrac{C}{2}\\&=\sin \dfrac{B+C}{2}\tan\dfrac{B}{2}\tan\dfrac{C}{2} \\&= \cos \dfrac{A}{2} \tan \dfrac{B}{2} \tan \dfrac{C}{2}\end{align*}$

and replacing this and similar identities for every term in the left hand side of our last inequality we have

$\displaystyle \sum_{cyc} \cos \dfrac{A}{2}\tan \dfrac{B}{2}\tan \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2}$

Multiplying this inequality by $\cot \dfrac{A}{2}\cot \dfrac{B}{2}\cot\dfrac{C}{2}=\cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot\dfrac{C}{2}$ we get

$\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$ and we are done.
 
Last edited:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top