Question Number 94628 by Vishal Sharma last updated on 20/May/20
$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{a}\:\bigtriangleup\:{ABC}\:\mathrm{is}\:\mathrm{6}\:\mathrm{times} \\ $$$$\mathrm{the}\:\mathrm{arithmetic}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sines}\:\mathrm{of} \\ $$$$\mathrm{its}\:\mathrm{angles}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{side}\:{a}\:\mathrm{is}\:\:\mathrm{1},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{angle}\:{A}\:\:\mathrm{is} \\ $$
Answered by Kunal12588 last updated on 20/May/20
![(a+b+c)=6×(1/3)(sin A + sin B +sin C) ((sin A)/a)=((sin B)/b)=((sin C)/c)=k [sine rule] (a+b+c)=2k(a+b+c) ⇒k=(1/2) ⇒sin A=(a/2) ⇒A=sin^(−1) (1/2) [∵ a=1] ⇒A=(π/6)=30°](Q94656.png)
$$\left({a}+{b}+{c}\right)=\mathrm{6}×\frac{\mathrm{1}}{\mathrm{3}}\left({sin}\:{A}\:+\:{sin}\:{B}\:+{sin}\:{C}\right) \\ $$$$\frac{{sin}\:{A}}{{a}}=\frac{{sin}\:{B}}{{b}}=\frac{{sin}\:{C}}{{c}}={k}\:\:\:\:\left[{sine}\:{rule}\right] \\ $$$$\left({a}+{b}+{c}\right)=\mathrm{2}{k}\left({a}+{b}+{c}\right) \\ $$$$\Rightarrow{k}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow{sin}\:{A}=\frac{{a}}{\mathrm{2}} \\ $$$$\Rightarrow{A}={sin}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\left[\because\:{a}=\mathrm{1}\right] \\ $$$$\Rightarrow{A}=\frac{\pi}{\mathrm{6}}=\mathrm{30}° \\ $$