Question Number 88752 by wiWiw last updated on 12/Apr/20
$$\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\alpha}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\beta}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\gamma}\right)\leqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{prove}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{inequality}} \\ $$
Commented by TANMAY PANACEA. last updated on 12/Apr/20
Commented by TANMAY PANACEA. last updated on 12/Apr/20
$${another}\:{approach} \\ $$$${point}\:{A},{B}\:{and}\:{C}\:{lies}\:{on}\:{curve}\:{y}={cosx} \\ $$$${point}\:{A}=\left({A},{cosA}\right) \\ $$$${point}\:{B}=\left({B},{cosB}\right) \\ $$$${point}\:{C}=\left({C},{cosC}\right) \\ $$$${centroid}\:{of}\:{triangle}\:{ABC}\:{is}\:{Q} \\ $$$${Q}=\left(\frac{{A}+{B}+{C}}{\mathrm{3}},\frac{{cosA}+{cosB}+{cosC}}{\mathrm{3}}\right) \\ $$$${from}\:{figure}\:\:{OP}=\frac{{A}+{B}+{C}}{\mathrm{3}} \\ $$$${PQ}=\frac{{cosA}+{cosB}+{cosC}}{\mathrm{3}} \\ $$$${Point}\:{R}\:{lies}\:{on}\:{y}={cosx} \\ $$$${co}\:{ordinate}\:{of}\:{point}\:{R}\:\left\{\frac{{A}+{B}+{C}}{\mathrm{3}},{cos}\left(\frac{{A}+{B}+{C}}{\mathrm{3}}\right)\right\} \\ $$$${from}\:{figure}\:{it}\:{is}\:{clear} \\ $$$${PR}>{PQ} \\ $$$${cos}\left(\frac{{A}+{B}+{C}}{\mathrm{3}}\right)>\frac{{cosA}+{cosB}+{cosC}}{\mathrm{3}} \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}>{cosA}+{cosB}+{cosC} \\ $$$$ \\ $$$$ \\ $$
Answered by mind is power last updated on 12/Apr/20
![we can assum a+b+c=π,a,b,c∈[0,(π/2)[ cos(x)′′=−cos(x)<0, concav ⇒(1/3)[cos(a)+cos(b)+cos(c)]≤cos(((a+b+c)/3))=cos((π/3)) ⇒cos(a)+coz(b)+cos(c)≤(3/2)](https://www.tinkutara.com/question/Q88755.png)
$${we}\:{can}\:{assum}\:{a}+{b}+{c}=\pi,{a},{b},{c}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\right.\right. \\ $$$${cos}\left({x}\right)''=−{cos}\left({x}\right)<\mathrm{0}, \\ $$$${concav} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{3}}\left[{cos}\left({a}\right)+{cos}\left({b}\right)+{cos}\left({c}\right)\right]\leqslant{cos}\left(\frac{{a}+{b}+{c}}{\mathrm{3}}\right)={cos}\left(\frac{\pi}{\mathrm{3}}\right) \\ $$$$\Rightarrow{cos}\left({a}\right)+{coz}\left({b}\right)+{cos}\left({c}\right)\leqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$ \\ $$