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) $\frac { 5 } { 4 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 5 } { 16 }$

Question

Quizplus · Accepted Answer

Given $\sinh x = \frac{3}{4}$, we need to find $\operatorname{coth} x$. The coth function is defined as $\operatorname{coth} x = \frac{\cosh x}{\sinh x}$. First, we need to find $\cosh x$. Using the identity $\cosh^2 x - \sinh^2 x = 1$, we can solve for $\cosh x$. Substituting $\sinh x = \frac{3}{4}$ into the identity gives us $\cosh^2 x - \left(\frac{3}{4}\right)^2 = 1$, which simplifies to $\cosh^2 x = 1 + \frac{9}{16} = \frac{25}{16}$. Taking the square root gives $\cosh x = \frac{5}{4}$ (choosing the positive root because $\cosh x$ is always positive for real $x$). Now, $\operatorname{coth} x = \frac{\cosh x}{\sinh x} = \frac{5/4}{3/4} = \frac{5}{3}$.

A Value of Sinh X or Cosh X Is Given sinh⁡x=34,coth⁡x=\sinh x = \frac { 3 } { 4 } , \operatorname { coth } x =sinhx=43​,cothx=

A Value of Sinh X or Cosh X Is Given $\sinh x = \frac { 3 } { 4 } , \operatorname { coth } x =$