Hey it’s Professor Dave, let’s learn about hyperbolic functions.
We’ve learned enough math up to this point that many of the mysterious buttons on a sophisticated
calculator have been elucidated.
There are trig functions, inverse trig functions, logarithms, exponential functions, and lots
of other things that are now well understood.
But on certain calculators you may also see these symbols, which look like sine, cosine,
and tangent, but with the letter H tacked on at the end.
What are these things?
These are called hyperbolic functions.
They are kind of like the trigonometric functions, except whereas the trig functions are related
to the unit circle, hyperbolic functions are related to the hyperbola.
Hopefully we remember what a hyperbola is.
The hyperbola is one of the conic sections we learned about, the tricky looking one where
the cross section goes from the base of one cone through to the base of the other, making
a shape kind of like a double parabola.
So how do hyperbolic functions relate to the hyperbola?
Well let’s look at hyperbolic sine X.
This is equal to (E to the X minus E to the negative X) over two.
Now what does this mean graphically?
Well let’s take this expression and break it up into two terms.
One half e to the X, minus one half e to the negative X.
If we sketch these two curves, we get a hyperbola, and the sum of these two curves at every point
will give us hyperbolic sine, which we can see crossing the origin and then getting closer
and closer to either section as we go to positive and negative infinity.
The domain and range are both all real numbers.
So from this we can see how hyperbolic sine, which can be expressed with these exponential
terms, is intimately linked with this hyperbola.
What about hyperbolic cosine?
That will be the same as hyperbolic sine, we just switch this minus to a plus.
That means that the part of the hyperbola that was below the X axis is now above it,
and if we add up these curves, we get hyperbolic cosine, which will look a bit different.
This one has a domain of all real numbers, but the range will be from one to positive infinity.
Hyperbolic tangent will just be hyperbolic sine over hyperbolic cosine, and that will
look like this, with horizontal asymptotes at negative one and one.
And then we can have hyperbolic versions of cosecant, secant, and cotangent, which are
just the reciprocals of these first three, as we might expect from our knowledge of trigonometric
functions.
What can we do with these hyperbolic functions?
Well there are a number of identities to be aware of, and they will actually look quite
familiar if we remember our trigonometric identities, from expressions with squared
functions, to these two sum formulas.
Amazingly, even the derivatives of hyperbolic functions will be very predictable, if we
remember the derivatives of trigonometric functions.
Here they are, with very few discrepancies from their trigonometric counterparts, just
a negative sign here or there.
This isn’t too surprising if we look at hyperbolic sine and differentiate.
We recall that the derivative of E to the X is E to the X, and so the only thing that
will change is that for the second term we use the chain rule, and bring the negative
sign from the exponent down here to cancel out the other one, to give us a plus sign.
That makes this expression equal to hyperbolic cosine, and all the other derivatives can
be found this way as well.
Now just as there are inverse trigonometric functions, there are inverse hyperbolic functions
as well.
These are denoted this way, and we can get their graphs by reflecting the regular hyperbolic
functions across the line y = x. Hyperbolic sine and tangent pass the horizontal line
test, meaning that their inverses pass the vertical line test and qualify as functions.
For hyperbolic cosine, the horizontal line test will fail, so we have to restrict the
domain to greater than or equal to zero, and from that we can get the inverse function.
If we want to express the inverse hyperbolic functions in terms of X, we can swap X and
Y, and then solve for Y, as we are used to.
See how we bring the two up here, then multiply through by E to the Y to get rid of that negative
exponent, since E to the Y times E to the negative Y is one.
Then we notice that this is actually in the form of a quadratic equation, since E to the
2Y is just (E to the Y) quantity squared.
If we use the quadratic formula, we end up with this expression for E to the Y, and then
to solve for Y we just take the natural log of both sides.
So here are the expressions for the three inverse hyperbolic functions.
These inverse hyperbolic functions are also differentiable, so let’s take the derivative
of inverse hyperbolic sine.
Given the composite function, we need the chain rule.
The derivative of natural log requires getting the reciprocal of what the log is operating
on, and then multiplying by the derivative of what was inside, and for that, this X becomes
one, and then the radical term requires another chain rule.
One half this term to the negative one half, times the derivative of what’s inside, which
is two X, and we get this.
Now we just change the number one to get a common denominator, so that we can combine
the two terms in the parenthetical, and then we can combine these two larger terms.
We find that this more complicated term will simply cancel out, and this is what we are
left with.
Doing similar work will get us the rest of these derivatives for the other inverse hyperbolic
functions, which are shown here.
So that covers the basics regarding hyperbolic functions.
We now understand the relationship these have with the hyperbola, and how to express them
using these exponential terms.
We can also take their derivatives, their inverses, and the derivatives of their inverses.
Now let’s move on to some other topics.