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Question Number 209880 by mnjuly1970 last updated on 24/Jul/24

Answered by mahdipoor last updated on 25/Jul/24

$$get:a=cteandA,B,Cisvariable$$$$f(A,B,C)=\frac{\sqrt{cosA}}{a}+\frac{\sqrt{cosB}}{b}+\frac{\sqrt{cosC}}{c}=$$$$\frac{1}{a}(\sqrt{cosA}+sinA(\frac{\sqrt{cosB}}{sinB}+\frac{\sqrt{cosC}}{sinC}))$$$$g(A,B,C)=A+B+C=180$$$$\Rightarrow \frac{\partial f}{\partial x_i}=\lambda \frac{\partial g}{\partial x_i}\Rightarrow f_{min}=f(60,60,60)=\frac{3\sqrt{cos60}}{a}=\frac{2.12}{a}$$$$,,$$$$h(A,B,C)=\frac{3}{a+b+c}=\frac{3sinA}{a(sinA+sinB+sinC)}$$$$\Rightarrow \frac{\partial h}{\partial x_i}=\lambda \frac{\partial g}{\partial x_i}\Rightarrow h_{max}=f(90,90,0)=\frac{1.5}{a}$$$$,,$$$$f⩾\frac{2.12}{a}>\frac{1.5}{a}⩾h\Rightarrow f⩾h\Rightarrow$$$$\frac{\sqrt{cosA}}{a}+\frac{\sqrt{cosB}}{b}+\frac{\sqrt{cosC}}{c}>\frac{3}{a+b+c}$$

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