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Calculate sinh x and cosh x given tanh x = 5/13

by RoRi

Find the values of \sinh x and \cosh x given that \tanh x = \frac{5}{13}.

Since \tanh^2 + \sech^2 = 1 (see here, Section 6.19 Exercise #14) we have

    \begin{align*} \tanh^2 x + \operatorname{sech}^2 x = 1 && \implies && \operatorname{sech}^2 x = \frac{144}{169} \\ && \implies && \operatorname{sech} x = \frac{12}{13} \\ && \implies && \cosh x = \frac{13}{12}. \end{align*}

Then, to compute \sinh x we have,

    \begin{align*} \tanh x = \frac{\sinh x}{\cosh x} && \implies && \frac{5}{13} &= \frac{12}{13} \sinh x \\ && \implies && \sinh x &= \frac{5}{12}. \end{align*}

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