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Question Number 92225 by  M±th+et+s last updated on 05/May/20

$$iftanh\left(x\right)=\frac{72}{161}\sqrt{5}$$$$provethatsinh\left(x\right)\in Q$$$$$$$$$$$$Q=\left\{rationalnumbdrs\right\}$$$$$$

Answered by MJS last updated on 05/May/20

$$y=\tanh x=\frac{{\mathrm{e}}^{2x}-1}{{\mathrm{e}}^{2x}+1}$$$$\Rightarrow x=\frac{1}{2}\ln \frac{1+y}{1-y}$$$$\sinh x=\frac{{\mathrm{e}}^x-{\mathrm{e}}^{-x}}{2}\wedge x=\frac{1}{2}\ln \frac{1+y}{1-y}$$$$\Rightarrow \sinh x=\frac{y}{\sqrt{1-y^2}}$$$$y=\frac{72}{161}\sqrt{5}$$$$\Rightarrow \sinh x=72\sqrt{5}\notin \mathbb{Q}\Rightarrow \mathrm{thesis}\mathrm{is}\mathrm{wrong}$$

Commented by  M±th+et+s last updated on 05/May/20

$$sorrysiritstanh\left(12x\right)itwasatypo$$

Commented by MJS last updated on 05/May/20

$$\mathrm{then}\mathrm{use}\mathrm{my}\mathrm{formulas}\mathrm{to}\mathrm{show}{\mathrm{it}}^{\prime }\mathrm{s}\frac{1}{2}$$$$\mathrm{btw}{\left(\frac{1+\sqrt{5}}{2}\right)}^3=2+\sqrt{5}$$

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