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.
-$\sinh x = \frac { 8 } { 15 } , \operatorname { sech } x =$
A) $\frac { 17 } { 15 }$
B) $\frac { 15 } { 8 }$
C) $\frac { 64 } { 289 }$
D) $\frac { 15 } { 17 }$

Question

Quizplus · Accepted Answer

Given $\sinh x = \frac{8}{15}$, we use the identity $\cosh^2 x - \sinh^2 x = 1$ to find $\cosh x$. First, solve for $\cosh x$: $\cosh^2 x = \sinh^2 x + 1 = \left(\frac{8}{15}\right)^2 + 1 = \frac{64}{225} + 1 = \frac{289}{225}$. Thus, $\cosh x = \sqrt{\frac{289}{225}} = \frac{17}{15}$. The $\operatorname{sech} x$ is the reciprocal of $\cosh x$, so $\operatorname{sech} x = \frac{1}{\cosh x} = \frac{15}{17}$.

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