## University Calculus: Early Transcendentals (3rd Edition)

$\sinh x=\dfrac{4}{3}\\ \cosh x= \dfrac{5}{3}\\ \tanh x= \dfrac{4}{5} \\ sech x=\dfrac{3}{5} \\ csch x=\dfrac{3}{4} \\ coth x=\dfrac{5}{4}$
We are given that: $\sinh x=\dfrac{4}{3}$ We need to find all the hyperbolic functions. Use identity: $\cosh^2 x-\sinh^2x=1$ or, $\cosh^2 x-(\dfrac{4}{3})^2=1 \implies \cosh^2 x=\dfrac{25}{9}$ and $\cosh x= \dfrac{5}{3}$ Now, $\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{4}{3}}{\dfrac{5}{3}}=\dfrac{4}{5}$ $sech x= \dfrac{1}{\cosh x}=\dfrac{1}{\dfrac{5}{3}}=\dfrac{3}{5}$; $csch x= \dfrac{1}{\sinh x}=\dfrac{1}{\dfrac{4}{3}}=\dfrac{3}{4}$; $coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{4}{5}}=\dfrac{5}{4}$