Question Number 38905 by Raj Singh last updated on 01/Jul/18
Answered by MrW3 last updated on 01/Jul/18
$$\frac{{a}}{\mathrm{sin}\:{A}}=\frac{{b}}{\mathrm{sin}\:{B}}=\frac{{c}}{\mathrm{sin}\:{C}}={k}\:\left({constant}\right) \\ $$$${let}\:{dA}\:={little}\:{change}\:{in}\:{angle}\:{A} \\ $$$$ \\ $$$${B}=\mathrm{180}−{C}−{A} \\ $$$$\Rightarrow{dB}=−{dA} \\ $$$$ \\ $$$${a}={k}\:\mathrm{sin}\:{A} \\ $$$${da}={k}\:\mathrm{cos}\:{A}\:{dA} \\ $$$$\Rightarrow\frac{{da}}{\mathrm{cos}\:{A}}={k}\:{dA} \\ $$$$ \\ $$$${b}={k}\:\mathrm{sin}\:{B} \\ $$$${db}={k}\:\mathrm{cos}\:{B}\:{dB}=−{k}\:\mathrm{cos}\:{B}\:{dA} \\ $$$$\Rightarrow\frac{{db}}{\mathrm{cos}\:{B}}=−{k}\:{dA} \\ $$$$ \\ $$$$\Rightarrow\frac{{da}}{\mathrm{cos}\:{A}}+\frac{{db}}{\mathrm{cos}\:{B}}={k}\:{dA}−{k}\:{dA}=\mathrm{0} \\ $$