Question

Hyperbolic functions

Answer 1

so sinh(x) is defined as
\[\sinh(x)=\frac{ e ^{x}-e ^{-x} }{ 2 }\]
\[\frac{ 5 }{ 4 }=\frac{ e ^{x}-e ^{-x} }{ 2 }\]
\[\ln(\frac{ 5 }{ 2 })=\ln(e ^{x}(1-e ^{-2x}))\]
\[\ln(\frac{ 5 }{ 2 })=xln(1-e ^{-2x})\]
need to think for a moment getting stuck on combining x's

Answer 2

\[\ln(\frac{ 5 }{ 2 })=x+\ln(1-e ^{-2x})\]

Answer 3

\[\sinh(x)=-isin(ix)\]

Answer 4

\[\ln(\frac{ 5 }{ 2 })=x+\ln(\frac{ e ^{2x} }{ e ^{2x} }-\frac{ 1 }{ e ^{2x} })\]
\[\ln(\frac{ 5 }{ 2 })=x+\ln(\frac{ e ^{2x}-1 }{ e ^{2x} })\]
\[\ln(\frac{ 5 }{ 2 })=x+\ln(e ^{2x}-1)-2x\]
\[\ln (\frac{ 5 }{ 2 })=\ln(e ^{2x}-1)-x\]
i'm trying to get x isolated may need to think hard about this one

Answer 5

\[asinh(x)=\ln(x+\sqrt{x+1})\]
\[x=asinh(\frac{ 5 }{ 4 })\]
\[x=\ln(\frac{ 5 }{ 4 }+\sqrt{\frac{ 5 }{ 4 }+1})\]
from the equation x~1

Answer 6

http://upload.wikimedia.org/math/b/e/f/bef508326a9b87db9f897769366ef1b5.png