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Question Number 175956 by mnjuly1970 last updated on 10/Sep/22

Commented by mnjuly1970 last updated on 10/Sep/22

prove ⇈⇈

$${prove}\:\upuparrows\upuparrows \\ $$

Answered by behi834171 last updated on 10/Sep/22

sin(A/2).sin(B/2).sin(C/2)≤sin^3 ((π/3)/2)=(1/8)  ⇒Σ(1/(sin^2 (A/2)))≥3×((Π(1/(sin^2 (A/2)))))^(1/3) ≥     ≥3×(1/( ((1/(64)))^(1/3) ))=3×4=12   .■

$${sin}\frac{{A}}{\mathrm{2}}.{sin}\frac{{B}}{\mathrm{2}}.{sin}\frac{{C}}{\mathrm{2}}\leqslant{sin}^{\mathrm{3}} \frac{\frac{\pi}{\mathrm{3}}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{8}} \\ $$$$\Rightarrow\Sigma\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \frac{{A}}{\mathrm{2}}}\geqslant\mathrm{3}×\sqrt[{\mathrm{3}}]{\Pi\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \frac{{A}}{\mathrm{2}}}}\geqslant \\ $$$$\:\:\:\geqslant\mathrm{3}×\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{64}}}}=\mathrm{3}×\mathrm{4}=\mathrm{12}\:\:\:.\blacksquare \\ $$

Commented by mnjuly1970 last updated on 10/Sep/22

mamnoon ostad  faghat : sin((A/2))sin((B/2))sin((C/2))≤(1/8)

$${mamnoon}\:{ostad} \\ $$$${faghat}\::\:{sin}\left(\frac{{A}}{\mathrm{2}}\right){sin}\left(\frac{{B}}{\mathrm{2}}\right){sin}\left(\frac{{C}}{\mathrm{2}}\right)\leqslant\frac{\mathrm{1}}{\mathrm{8}} \\ $$

Commented by mnjuly1970 last updated on 10/Sep/22

zendeh bashid

$${zendeh}\:{bashid} \\ $$

Commented by behi834171 last updated on 10/Sep/22

3pas ostad.eslah shod.

$$\mathrm{3}{pas}\:{ostad}.{eslah}\:{shod}. \\ $$

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