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Question Number 144662 by mnjuly1970 last updated on 27/Jun/21

$$...........\mathrm{Calculus}...........$$$$\mathrm{In}\mathrm{A}\stackrel{\mathrm{\Delta }}{\mathrm{B}C}:$$$$\hat{\mathrm{B}}=2\hat{\mathrm{C}},a=\lambda bthenspecify$$$$thelimitsofthechanges{}^{\prime }\lambda {}^{\prime }:$$$$$$$$$$

Answered by mindispower last updated on 27/Jun/21

$$\frac{a}{b}=\lambda$$$$\frac{a}{sin\left(A\right)}=\frac{b}{sin\left(B\right)}=\frac{c}{sin\left(C\right)}$$$$\lambda =\frac{sin(\pi -\left(3c\right))}{sin\left(2c\right)}=\frac{sin\left(3c\right)}{sin\left(c\right)}$$$$sin\left(3c\right)=3sin\left(c\right)-4{sin}^2\left(c\right)$$$$\lambda =3-4{sin}^2\left(c\right)$$$$A+B+C=\pi$$$$A+3C=\pi$$$$A=\pi -3C>0\Rightarrow 0<C<\frac{\pi }{3}$$$${sin}^2\left(C\right)⩽{sin}^2\left(\frac{\pi }{3}\right)=\frac{3}{4}$$$$\lambda ⩾3-\frac{4.3}{4}=0$$$$\lambda \in ]0,3[$$

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