Answer
$\sinh x=\dfrac{4}{3}\\ \cosh x= \dfrac{5}{3}\\ \tanh x= \dfrac{4}{5} \\ sech x=\dfrac{3}{5} \\ csch x=\dfrac{3}{4} \\ coth x=\dfrac{5}{4}$
Work Step by Step
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, $\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{4}{3}}{\dfrac{5}{3}}=\dfrac{4}{5}$
$sech x= \dfrac{1}{\cosh x}=\dfrac{1}{\dfrac{5}{3}}=\dfrac{3}{5}$;
$csch x= \dfrac{1}{\sinh x}=\dfrac{1}{\dfrac{4}{3}}=\dfrac{3}{4}$;
$coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{4}{5}}=\dfrac{5}{4}$
