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Given the value of sinh(x)

for example sinh(x) = 3/2

How can I calculate the value of cosh(x) ?

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9  
Use identity: $\cosh^2(x)-\sinh^2(x)=1$ –  alans Jul 11 at 18:47
4  
@alans When you hover your mouse over "add comment" you will see the following: Use comments to ask for more information or to suggest improvements. Avoid answering questions in comments. –  Fly by Night Jul 11 at 19:26

2 Answers 2

up vote 9 down vote accepted

Use the identity $\cosh^2x-\sinh^2x \equiv 1$. If $\sinh x = \frac{3}{2}$ then $$\cosh^2x - \left(\frac{3}{2}\right)^{\! 2} = 1$$ $$\cosh^2x - \frac{9}{4} = 1$$ $$\cosh^2x = \frac{13}{4}$$ It follows that $\cosh x = \pm\frac{1}{2}\sqrt{13}$. Since $\cosh x \ge 1$ for all $x \in \mathbb{R}$ we have $\cosh x = \frac{1}{2}\sqrt{13}$.

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Please explain the downvote. –  Fly by Night Jul 11 at 18:51
    
You made a sign error. $\sinh$ maps $\mathbb{R}$ onto itself. –  Ian Jul 11 at 18:51
    
I think you want $\sinh x = \frac{3}{2}$ in your premise, not $\sinh^2$. (I tried to edit but it wasn't substantial enough since it was only two characters.) –  Ismail Badawi Jul 11 at 20:11
    
@IsmailBadawi You should not edit other people's posts. –  Fly by Night Jul 11 at 20:52
6  
@FlybyNight Stack Exchange disagrees with you: meta.stackexchange.com/questions/120576/… –  Ismail Badawi Jul 11 at 21:15

The trick is....

$$\cosh^2x-\sinh^2x=1$$

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