A value of $\sinh x$ or $\cosh x$ is given. Use the definitions and the identity $\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$ to find the value of the other indicated hyperbolic function.
-$\sinh x = - \frac { 4 } { 3 } , \tanh x =$
A) $- \frac { 5 } { 4 }$
B) $\frac { 5 } { 3 }$
C) $- \frac { 4 } { 5 }$
D) $\frac { 4 } { 5 }$

Question

Quizplus · Accepted Answer

Given $\sinh x = -\frac{4}{3}$, we can find $\cosh x$ using the identity $\cosh^2 x - \sinh^2 x = 1$. Substituting $\sinh x = -\frac{4}{3}$ into the identity gives us $\cosh^2 x - \left(-\frac{4}{3}\right)^2 = 1$, which simplifies to $\cosh^2 x = 1 + \frac{16}{9}$, leading to $\cosh^2 x = \frac{25}{9}$. Taking the square root gives $\cosh x = \frac{5}{3}$ (choosing the positive root because $\cosh x$ is always positive). Then, $\tanh x = \frac{\sinh x}{\cosh x} = \frac{-\frac{4}{3}}{\frac{5}{3}} = -\frac{4}{5}$.

A Value Of sinh⁡x\sinh xsinhx Or cosh⁡x\cosh xcoshx Is Given Use the Definitions and the Identity

A Value Of $\sinh x$ Or $\cosh x$ Is Given Use the Definitions and the Identity