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Question Number 94628 by Vishal Sharma last updated on 20/May/20

$$\mathrm{The}\mathrm{perimeter}\mathrm{of}\mathrm{a}△ABC\mathrm{is}6\mathrm{times}$$$$\mathrm{the}\mathrm{arithmetic}\mathrm{mean}\mathrm{of}\mathrm{the}\mathrm{sines}\mathrm{of}$$$$\mathrm{its}\mathrm{angles}.\mathrm{If}\mathrm{the}\mathrm{side}a\mathrm{is}1,\mathrm{then}\mathrm{the}$$$$\mathrm{angle}A\mathrm{is}$$

Answered by Kunal12588 last updated on 20/May/20

$$(a+b+c)=6\times \frac{1}{3}(sinA+sinB+sinC)$$$$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}=k\left[sinerule\right]$$$$(a+b+c)=2k(a+b+c)$$$$\Rightarrow k=\frac{1}{2}$$$$\Rightarrow sinA=\frac{a}{2}$$$$\Rightarrow A={sin}^{-1}\frac{1}{2}[\because a=1]$$$$\Rightarrow A=\frac{\pi }{6}=30°$$

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