MHB Proving an Inequality Involving Sines

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The inequality involving sines states that for a triangle with angles A, B, and C, and corresponding sides a, b, and c, the sum of half-angle sines is less than or equal to the sum of ratios of sides to the sum of the other two sides. The proof utilizes the Law of Sines to express the sides in terms of the angles. By applying sum-to-product identities and simplifications, the relationship between the sine values and the side ratios is established. The conclusion is reached by demonstrating that each sine term is bounded by its corresponding ratio, confirming the inequality. This establishes a fundamental relationship in triangle geometry.
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Prove that $\sin \frac{A}{2}+\sin\frac{B}{2}+\sin \frac{C}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$
 
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Re: TRig Inequality

jacks said:
Prove that $\sin \frac{A}{2}+\sin\frac{B}{2}+\sin \frac{C}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$

Hi jacks, :)

I hope \(A,\,B\mbox{ and } C\) are the internal angles of a triangle and \(a,\,b,\,c\) are the sides opposite to those angles respectively. In that case you can use the Law of sines.

\[\frac{a}{\sin A} \,=\, \frac{b}{\sin B} \,=\, \frac{c}{\sin C} \!\]

\begin{eqnarray}

\frac{b}{a+c}&=&\frac{1}{\frac{a}{b}+\frac{c}{b}}\\

&=&\frac{1}{\frac{\sin A}{\sin B}+\frac{\sin C}{\sin B}} \\

&=&\frac{\sin B}{\sin A+\sin C}\\

\end{eqnarray}

Similarly,

\[\frac{c}{a+b}=\frac{\sin C}{\sin A+\sin B}\mbox{ and }\frac{a}{b+c}=\frac{\sin A}{\sin B+\sin C}\]

\[\therefore\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{\sin B}{\sin A+\sin C}+\frac{\sin C}{\sin A+\sin B}+\frac{\sin A}{\sin B+\sin C}\]

Using the sum to product identities and simplification gives,

\[\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{ \sin\frac{B}{2}}{\cos\left(\frac{A-C}{2}\right)}+\frac{\sin\frac{A}{2}}{\cos\left( \frac{B-C}{2}\right)}+\frac{\sin\frac{C}{2}}{\cos\left( \frac{A-B}{2}\right)}~~~~~~~~~~~~~(1)\]

Note that, \(\displaystyle\left|\cos\left( \frac{B-C}{2}\right)\right|\leq 1\Rightarrow \frac{1}{\left|\cos\left( \frac{B-C}{2}\right)\right|}\geq 1\Rightarrow \frac{\sin \frac{A}{2}}{\left|\cos\left( \frac{B-C}{2}\right)\right|}\geq \sin \frac{A}{2}\Rightarrow -\sin \frac{A}{2}\geq\frac{\sin\frac{A}{2}}{\cos\left( \frac{B-C}{2}\right)}\mbox{ or }\frac{\sin\frac{A}{2}}{\cos\left( \frac{B-C}{2}\right)}\geq \sin \frac{A}{2}~~~~~~~~~~~~~(2)\)

Similarly, \[-\sin \frac{B}{2}\geq\frac{\sin\frac{B}{2}}{\cos\left( \frac{A-C}{2}\right)}\mbox{ or }\frac{\sin\frac{B}{2}}{\cos\left( \frac{A-C}{2}\right)}\geq \sin \frac{B}{2}~~~~~~~~~~~~~(3)\]

\[-\sin \frac{C}{2}\geq\frac{\sin\frac{C}{2}}{\cos\left( \frac{A-B}{2}\right)}\mbox{ or }\frac{\sin\frac{C}{2}}{\cos\left( \frac{A-B}{2}\right)}\geq \sin \frac{C}{2}~~~~~~~~~~~~~(4)\]

By (1), (2), (3) and (4),

\[\sin \frac{A}{2}+\sin\frac{B}{2}+\sin \frac{C}{2}\leq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\]

Kind Regards,
Sudharaka.
 
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