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Question Number 173983 by mnjuly1970 last updated on 23/Jul/22
in AB^Δ C prove : (( cos(A ))/a^( 3) ) +((cos(B))/b^( 3) ) +((cos(C))/c^( 3) ) ≥((81)/(16p^( 3) )) where : p= (a+b +c )/2
$$ \\ $$$$\:\:\:{in}\:{A}\overset{\Delta} {{B}C}\:\:{prove}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\frac{\:{cos}\left({A}\:\right)}{{a}^{\:\mathrm{3}} }\:+\frac{{cos}\left({B}\right)}{{b}^{\:\mathrm{3}} }\:+\frac{{cos}\left({C}\right)}{{c}^{\:\mathrm{3}} }\:\geqslant\frac{\mathrm{81}}{\mathrm{16}{p}^{\:\mathrm{3}} } \\ $$$$\:\:\:{where}\::\:\:{p}=\:\left({a}+{b}\:+{c}\:\right)/\mathrm{2} \\ $$$$ \\ $$
Answered by mnjuly1970 last updated on 23/Jul/22
proof : equivalently we must prove: ((cos(A))/(sin^( 3) (A))) +((cos(B))/(sin^( 3) (B))) +((cos(C))/(sin^( 3) (C))) ≥ ((81)/(2 {(sin(A)+sin(B)+sin(c))^3 ≥^? 27sin(A)sin(B)sin(C)})) also ((cos(A).sin(B).sin(C))/(sin^( 2) (A))) + ((cos(B).sin(A).sin(C))/(sin^( 2) (B))) + (( cos(C).sin(A).sin(B))/(sin^( 2) (C))) ≥(3/2) equivalently: ((cos(A).bc)/a^( 2) ) +((cos(B).ac)/b^( 2) )+((cos(C).ab)/c^( 2) )≥(3/2) also ((b^( 2) +c^( 2) −a^2 )/(2a^( 2) )) +((a^( 2) +c^( 2) −b^( 2) )/(2b^( 2) ))+((a^( 2) +b^( 2) −c^( 2) )/(2c^( 2) ))≥(3/2) equivalently: ∗{ ((b^( 2) +c^( 2) )/a^( 2) ) +((a^( 2) +c^( 2) )/b^( 2) ) +((a^( 2) +b^( 2) )/c^( 2) )} ≥^? 6 ∗ L.h.s ≥^(am−gm) (2(√((a^( 2) /b^( 2) ) .(b^( 2) /a^( 2) ))) +2(√((a^( 2) /c^( 2) ).(c^( 2) /a^( 2) ))) +2(√((b^( 2) /c^( 2) ).(c^( 2) /(b 2)) ))) L.h.s ≥ 6 ■
$$\:\:{proof}\:: \\ $$$$\:\:{equivalently}\:{we}\:{must}\:{prove}: \\ $$$$\:\:\:\:\frac{{cos}\left({A}\right)}{{sin}^{\:\mathrm{3}} \left({A}\right)}\:+\frac{{cos}\left({B}\right)}{{sin}^{\:\mathrm{3}} \left({B}\right)}\:+\frac{{cos}\left({C}\right)}{{sin}^{\:\mathrm{3}} \left({C}\right)} \\ $$$$\:\:\:\:\:\geqslant\:\frac{\mathrm{81}}{\mathrm{2}\:\left\{\left({sin}\left({A}\right)+{sin}\left({B}\right)+{sin}\left({c}\right)\right)^{\mathrm{3}} \overset{?} {\geqslant}\mathrm{27}{sin}\left({A}\right){sin}\left({B}\right){sin}\left({C}\right)\right\}} \\ $$$$\:\:\:\:{also} \\ $$$$\:\:\:\frac{{cos}\left({A}\right).{sin}\left({B}\right).{sin}\left({C}\right)}{{sin}^{\:\mathrm{2}} \left({A}\right)}\:+ \\ $$$$\:\:\:\:\frac{{cos}\left({B}\right).{sin}\left({A}\right).{sin}\left({C}\right)}{{sin}^{\:\mathrm{2}} \left({B}\right)}\:+ \\ $$$$\:\:\:\:\frac{\:{cos}\left({C}\right).{sin}\left({A}\right).{sin}\left({B}\right)}{{sin}^{\:\mathrm{2}} \left({C}\right)}\:\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\:\:\:\:{equivalently}: \\ $$$$\:\:\:\:\frac{{cos}\left({A}\right).{bc}}{{a}^{\:\mathrm{2}} }\:+\frac{{cos}\left({B}\right).{ac}}{{b}^{\:\mathrm{2}} }+\frac{{cos}\left({C}\right).{ab}}{{c}^{\:\mathrm{2}} }\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\:\:\:\: \\ $$$$\:\:\:{also} \\ $$$$\:\:\:\:\frac{{b}^{\:\mathrm{2}} +{c}^{\:\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{a}^{\:\mathrm{2}} }\:+\frac{{a}^{\:\mathrm{2}} +{c}^{\:\mathrm{2}} −{b}^{\:\mathrm{2}} }{\mathrm{2}{b}^{\:\mathrm{2}} }+\frac{{a}^{\:\mathrm{2}} +{b}^{\:\mathrm{2}} −{c}^{\:\mathrm{2}} }{\mathrm{2}{c}^{\:\mathrm{2}} }\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\:\:\:\:{equivalently}: \\ $$$$\:\:\:\ast\left\{\:\frac{{b}^{\:\mathrm{2}} +{c}^{\:\mathrm{2}} }{{a}^{\:\mathrm{2}} }\:+\frac{{a}^{\:\mathrm{2}} +{c}^{\:\mathrm{2}} }{{b}^{\:\mathrm{2}} }\:+\frac{{a}^{\:\mathrm{2}} +{b}^{\:\mathrm{2}} }{{c}^{\:\mathrm{2}} }\right\}\:\overset{?} {\geqslant}\mathrm{6} \\ $$$$\:\:\:\ast\:{L}.{h}.{s}\:\overset{\boldsymbol{{am}}−\boldsymbol{{gm}}} {\geqslant}\:\left(\mathrm{2}\sqrt{\frac{{a}^{\:\mathrm{2}} }{{b}^{\:\mathrm{2}} }\:.\frac{{b}^{\:\mathrm{2}} }{{a}^{\:\mathrm{2}} }}\:+\mathrm{2}\sqrt{\frac{{a}^{\:\mathrm{2}} }{{c}^{\:\mathrm{2}} }.\frac{{c}^{\:\mathrm{2}} }{{a}^{\:\mathrm{2}} }}\:+\mathrm{2}\sqrt{\left.\frac{{b}^{\:\mathrm{2}} }{{c}^{\:\mathrm{2}} }.\frac{{c}^{\:\mathrm{2}} }{{b}\:\mathrm{2}}\:\right)}\right. \\ $$$$\:\:\:\:\:\:\:\:{L}.{h}.{s}\:\geqslant\:\mathrm{6}\:\:\blacksquare \\ $$$$\:\:\:\:\: \\ $$$$\:\:\: \\ $$

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