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.
-$\sinh x = \frac { 3 } { 4 } , \operatorname { coth } x =$
A) $\frac { 3 } { 5 }$
B) $\frac { 5 } { 16 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 5 } { 4 }$

Question

Quizplus · Accepted Answer

Given $\sinh x = \frac{3}{4}$, we can find $\cosh x$ using the identity $\cosh^2 x - \sinh^2 x = 1$. Solving for $\cosh x$, we get $\cosh x = \sqrt{\sinh^2 x + 1} = \sqrt{\left(\frac{3}{4}\right)^2 + 1} = \sqrt{\frac{9}{16} + 1} = \sqrt{\frac{25}{16}} = \frac{5}{4}$. The coth function is defined as $\coth x = \frac{\cosh x}{\sinh x}$, so substituting the values we get $\coth x = \frac{5/4}{3/4} = \frac{5}{3}$.

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