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Inverse Hyperbolic Trigonometric Functions

For background on inverse functions, refer to a previous post on the inverse standard trig functions. Looking at the graphs of the hyperbolic trig functions in the previous post, we can see that sinh, tanh, csch, and coth are one-to-one, and therefore have a well-defined inverse. By contrast, cosh and sech are not one-to-one, so we will have to restrict their domain. Since we have exact formulas for the hyperbolic trig functions in terms of ex, we will also get exact formulas for their inverse functions in terms of lnx.


Derivations

Hyperbolic Sine

The standard method for solving for inverse functions is to swap x and y and then isolate for y.

y=sinh(x)=exex2x=eyey2

Now, we are going to isolate ey, which will then let us isolate for y. Multiply both sides by 2ey.

2xey=e2y1(ey)22xey1=0

Use the quadratic formula.

ey=2x±4x2+42ey=x±x2+1

Since x<x2+1, the root xx2+1 will result in a negative number and thus no solution for y.

ey=x+x2+1y=arcsinh(x)=ln(x+x2+1)

Hyperbolic Cosine

The steps are almost identical to arcsinh. I will skip a few steps to save space.

y=cosh(x)=ex+ex2x=ey+ey2(ey)22xey+1=0ey=x±x21ey=x+x21y=arccosh(x)=ln(x+x21)

One thing to note is that the reason we neglect the root ey=xx21 is different than before. x>x21, therefore the root will be positive and produce a solution for y. Our choice of using the root ey=x+x21 is due to our choice of domain restriction.

Hyperbolic Tangent

The methodology is very similar to the above. Again, I will skip some steps to save space.

y=tanh(x)=exexex+exx=eyeyey+ey=e2y1e2y+1e2y=1+x1xy=arctanh(x)=12ln(1+x1x)

Hyperbolic Secant

This is almost identical to the derivation of arccosh.

y=sech(x)=2ex+exx=2ex+ex(ey)22xey+1=0ey=1x±1x21ey=1x+1x21y=arcsech(x)=ln(1x+1x21 )

Hyperbolic Cosecant

This is almost identical to the derivation of arcsinh.

y=csch(x)=2exexx=2exex(ey)2+2xey+1=0ey=1x±1x2+1ey=1x+1x2+1y=arccsch(x)=ln(1x+1x2+1 )

Hyperbolic Cotangent

This is almost identical to the derivation of arctanh.

y=coth(x)=ex+exexexx=ey+eyeyey=e2y+1e2y1e2y=x+1x1y=arccoth(x)=12ln(x+1x1)


Summary

arcsinh(x)sinh1(x)=ln(x+x2+1)arccosh(x)(cosh|[0,))1(x)=ln(x+x21)arctanh(x)tanh1(x)=12ln(1+x1x)arcsech(x)(sech|[0,))1(x)=ln(1x+1x21)arccsch(x)(csch|(,0)(0,))1(x)=ln(1x+1x2+1)arccoth(x)(coth|(,0)(0,))1(x)=12ln(1x1+x)

Trigonometry Series