tgt

## Wednesday, 30 July 2014

### CHAPTER 11 - Inverse Hyperbolic Functions

Inverse Hyperbolic Functions

The hyperbolic sine function is a one-to-one function, and thus has an inverse. As usual, we obtain the graph of the inverse hyperbolic sine function  (also denoted by  ) by reflecting the graph of about the line y=x:

Since  is defined in terms of the exponential function, you should not be surprised that its inverse function can be expressed in terms of the logarithmic function:
Let's set  , and try to solve for x:

This is a quadratic equation with  instead of x as the variable. y will be considered a constant.
So using the quadratic formula, we obtain

Since  for all x, and since  for all y, we have to discard the solution with the minus sign, so

and consequently

Read that last sentence again slowly!
We have found out that

#### Try it yourself!

You know what's coming up, don't you? Here's the graph. Note that the hyperbolic cosine function is not one-to-one, so let's restrict the domain to  .

Here it is: Express the inverse hyperbolic cosine functions in terms of the logarithmic function!

# Hyperbolic Functions

The hyperbolic functions enjoy properties similar to the trigonometric functions; their definitions, though, are much more straightforward:

Here are their graphs: the  (pronounce: "kosh") is pictured in red, the  function (rhymes with the "Grinch") is depicted in blue.

As their trigonometric counterparts, the  function is even, while the  function is odd.
Their most important property is their version of the Pythagorean Theorem.

The verification is straightforward:
While  ,  , parametrizes the unit circle, the hyperbolic functions  ,  , parametrize the standard hyperbola  , x>1.
In the picture below, the standard hyperbola is depicted in red, while the point  for various values of the parameter t is pictured in blue.

The other hyperbolic functions are defined the same way, the rest of the trigonometric functions is defined:

 tanh x coth x sech x csch x

For every formula for the trigonometric functions, there is a similar (not necessary identical) formula for the hyperbolic functions:
Let's consider for example the addition formula for the hyperbolic cosine function:

#### Try it yourself!

Prove the addition formula for the hyperbolic sine function:Show that  .

## The Derivatives of Trigonometric Functions

Trigonometric functions are useful in our practical lives in diverse areas such as astronomy, physics, surveying, carpentry etc. How can we find the derivatives of the trigonometric functions?
Our starting point is the following limit:

Using the derivative language, this limit means that . This limit may also be used to give a related one which is of equal importance:

To see why, it is enough to rewrite the expression involving the cosine as

But , so we have

This limit equals  and thus .

In fact, we may use these limits to find the derivative of  and  at any point x=a. Indeed, using the addition formula for the sine function, we have

So

which implies

So we have proved that  exists and .Similarly, we obtain that  exists and that .
Since , and  are all quotients of the functions  and , we can compute their derivatives with the help of the quotient rule:

It is quite interesting to see the close relationship between  and  (and also between  and ).
From the above results we get

These two results are very useful in solving some differential equations.Example 1. Let . Using the double angle formula for the sine function, we can rewrite

So using the product rule, we get

which implies, using trigonometric identities,

Exercise 1. Find the equations of the tangent line and the normal line to the graph of  at the point .
Answer. First we need to find the derivative of f(x) so we can get the slope of the tangent line and the normal line. We have

So we have

knowing that  and . Note that

So the equation of the tangent line at the point  is

the slope of the normal line to the graph at the point  is

which helps us find the equation of the normal line as

Exercise 2. Find the x-coordinates of all points on the graph of  in the interval  at which the tangent line is horizontal.
Answer. The points (x,f(x)) at which the tangent line is horizontal are the ones for which f'(x) = 0. Let us first find f'(x). We have

So we have to solve  which gives . We get