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solve-tan-x-tan-2x-2-3-




Question Number 96766 by bemath last updated on 04/Jun/20
solve : tan x−tan (2x) = 2(√3)
$$\mathrm{solve}\::\:\mathrm{tan}\:{x}−\mathrm{tan}\:\left(\mathrm{2}{x}\right)\:=\:\mathrm{2}\sqrt{\mathrm{3}}\: \\ $$
Answered by bobhans last updated on 04/Jun/20
⇒ tan (x)−((2tan (x))/(1−tan^2 (x))) = 2(√3)  tan (x)−tan^3 (x)−2tan (x)= 2(√3) −2(√3) tan^2 (x)  let tan (x) = v ⇒v^3 −2(√3) v^2 + v+ 2(√3) = 0  (v−(√3)) (v^2 −(√3)v −2) = 0   { ((v = (√3) )),((v = (((√3) + (√(11)))/2))),((v = (((√3) −(√(11)))/2))) :}   { ((tan (x) = (√3) ⇒x = (π/3)+ nπ)),((tan (x) = (((√3)+(√(11)))/2) ⇒ x = arctan ((((√3)+(√(11)))/2))+nπ)),((tan (x) = (((√3)−(√(11)))/2) ⇒x = arctan ((((√3)−(√(11)))/2)) +nπ)) :}
$$\Rightarrow\:\mathrm{tan}\:\left({x}\right)−\frac{\mathrm{2tan}\:\left({x}\right)}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}\:=\:\mathrm{2}\sqrt{\mathrm{3}} \\ $$$$\mathrm{tan}\:\left({x}\right)−\mathrm{tan}\:^{\mathrm{3}} \left({x}\right)−\mathrm{2tan}\:\left({x}\right)=\:\mathrm{2}\sqrt{\mathrm{3}}\:−\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{tan}\:^{\mathrm{2}} \left({x}\right) \\ $$$${let}\:\mathrm{tan}\:\left({x}\right)\:=\:{v}\:\Rightarrow{v}^{\mathrm{3}} −\mathrm{2}\sqrt{\mathrm{3}}\:{v}^{\mathrm{2}} +\:{v}+\:\mathrm{2}\sqrt{\mathrm{3}}\:=\:\mathrm{0} \\ $$$$\left({v}−\sqrt{\mathrm{3}}\right)\:\left({v}^{\mathrm{2}} −\sqrt{\mathrm{3}}{v}\:−\mathrm{2}\right)\:=\:\mathrm{0} \\ $$$$\begin{cases}{{v}\:=\:\sqrt{\mathrm{3}}\:}\\{{v}\:=\:\frac{\sqrt{\mathrm{3}}\:+\:\sqrt{\mathrm{11}}}{\mathrm{2}}}\\{{v}\:=\:\frac{\sqrt{\mathrm{3}}\:−\sqrt{\mathrm{11}}}{\mathrm{2}}}\end{cases} \\ $$$$\begin{cases}{\mathrm{tan}\:\left({x}\right)\:=\:\sqrt{\mathrm{3}}\:\Rightarrow{x}\:=\:\frac{\pi}{\mathrm{3}}+\:\mathrm{n}\pi}\\{\mathrm{tan}\:\left({x}\right)\:=\:\frac{\sqrt{\mathrm{3}}+\sqrt{\mathrm{11}}}{\mathrm{2}}\:\Rightarrow\:{x}\:=\:\mathrm{arctan}\:\left(\frac{\sqrt{\mathrm{3}}+\sqrt{\mathrm{11}}}{\mathrm{2}}\right)+{n}\pi}\\{\mathrm{tan}\:\left({x}\right)\:=\:\frac{\sqrt{\mathrm{3}}−\sqrt{\mathrm{11}}}{\mathrm{2}}\:\Rightarrow{x}\:=\:\mathrm{arctan}\:\left(\frac{\sqrt{\mathrm{3}}−\sqrt{\mathrm{11}}}{\mathrm{2}}\right)\:+{n}\pi}\end{cases} \\ $$$$ \\ $$

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