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Question Number 130806 by shaker last updated on 29/Jan/21

Answered by Olaf last updated on 29/Jan/21

shx = (2/3) = ((e^x −e^(−x) )/2)  e^(2x) −(4/3)e^x −1 = 0  e^x  = (1/3)(2+(√(13)))  e^(−x)  = (3/(2+(√(13)))) = (1/3)(2−(√(13)))  e^(3x)  = (1/(27))(2+(√(13)))^3  = (1/(27))(86+25(√(13)))  e^(−3x)  = (1/(27))(2−(√(13)))^3  = (1/(27))(86−25(√(13)))  coth(3x) = ((e^(3x) +e^(−3x) )/(e^(3x) −e^(−3x) ))  coth(3x) = ((86)/(25(√(13))))

$$\mathrm{sh}{x}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\:=\:\frac{{e}^{{x}} −{e}^{−{x}} }{\mathrm{2}} \\ $$$${e}^{\mathrm{2}{x}} −\frac{\mathrm{4}}{\mathrm{3}}{e}^{{x}} −\mathrm{1}\:=\:\mathrm{0} \\ $$$${e}^{{x}} \:=\:\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{2}+\sqrt{\mathrm{13}}\right) \\ $$$${e}^{−{x}} \:=\:\frac{\mathrm{3}}{\mathrm{2}+\sqrt{\mathrm{13}}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{2}−\sqrt{\mathrm{13}}\right) \\ $$$${e}^{\mathrm{3}{x}} \:=\:\frac{\mathrm{1}}{\mathrm{27}}\left(\mathrm{2}+\sqrt{\mathrm{13}}\right)^{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{27}}\left(\mathrm{86}+\mathrm{25}\sqrt{\mathrm{13}}\right) \\ $$$${e}^{−\mathrm{3}{x}} \:=\:\frac{\mathrm{1}}{\mathrm{27}}\left(\mathrm{2}−\sqrt{\mathrm{13}}\right)^{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{27}}\left(\mathrm{86}−\mathrm{25}\sqrt{\mathrm{13}}\right) \\ $$$$\mathrm{coth}\left(\mathrm{3}{x}\right)\:=\:\frac{{e}^{\mathrm{3}{x}} +{e}^{−\mathrm{3}{x}} }{{e}^{\mathrm{3}{x}} −{e}^{−\mathrm{3}{x}} } \\ $$$$\mathrm{coth}\left(\mathrm{3}{x}\right)\:=\:\frac{\mathrm{86}}{\mathrm{25}\sqrt{\mathrm{13}}} \\ $$

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