Answer
$\sinh x=\dfrac{-3}{4}\\ \cosh x= \dfrac{5}{4}\\ \tanh x= \dfrac{-3}{5} \\ sech x=\dfrac{4}{5} \\ csch x=\dfrac{-4}{3}$ and $ coth x=\dfrac{-5}{3}$
Work Step by Step
Given: $\sinh x=\dfrac{-3}{4}$
The remaining hyperbolic functions can be found as follows:
Use the fact $\cosh^2 x-\sinh^2x=1$
or, $\cosh^2 x-(\dfrac{-3}{4})^2=1 \\ or, \cosh^2 x=\dfrac{25}{16} $
Thus, $\cosh x= \dfrac{5}{4} \\
\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{-3}{4}}{\dfrac{5}{4}}=\dfrac{-3}{5} and \\
sech x= \dfrac{1}{\cosh x}=\dfrac{1}{\dfrac{5}{4}}=\dfrac{4}{5} \\
csch x= \dfrac{1}{\sinh x}=\dfrac{1}{\dfrac{-3}{4}}=\dfrac{-4}{3} \\
coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{-3}{5}}=\dfrac{-5}{3}$
