Answer
$\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} $ and $ coth x=\dfrac{5}{4}$
Work Step by Step
Given: $\sinh x=\dfrac{4}{3}$
The remaining hyperbolic functions can be found as follows:
Use the fact $\cosh^2 x-\sinh^2x=1$
or, $\cosh^2 x-(\dfrac{4}{3})^2=1 \\ or, \cosh^2 x=\dfrac{25}{9} $
Thus, $\cosh x= \dfrac{5}{3} \\
\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{4}{3}}{\dfrac{5}{3}}=\dfrac{4}{5} $
Now, $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} $
and $ coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{4}{5}}=\dfrac{5}{4}$
