Hyperbolic functions are the rare cousins of the trigonometry functions. While one might think that these functions deal solely with angles, they are actually combinations of exponential functions and their uses can be seen in various applications.
A high school course will most probably not teach hyperbolic functions because their identities and derivatives, while similar to but not entirely like that of trigonometry, can cause confusion. Nonetheless, we will teach some of it here are Gaussian Math, just enough to solve the catenary problem.
HYPERBOLIC FUNCTIONS.1
May want to check out:
INTEGRATING BY PARTS
HYPERBOLIC FUNCTIONS.2
DERIVATIVES OF HYPERBOLIC FUNCTIONS
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VIDEO LECTURE
part 1
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LESSON
The two basic hyperbolic functions are the hyperbolic sine and hyperbolic cosine defined by
The “sinh” function is pronounced as “cinch” rhyming with pinch while the “cosh” function is pronounced as “cosh” rhyming with gosh. Similar to trigonometry, these two functions have the following properties,
I like to say that the cosh function can absorb the ‘minus’ sign. The four identities you should be familiar with are
While the addition formula for hyperbolic functions are analogous to that of trigonometry for the sinh function, notice the change of sign for the cosh function. Most of these identities can be proven using the definition for the start. one such example is
Here is a quick comparison of trigonometry and hyperbolic functions as used in rectangular equations of graph. Using parametric equations, the point
Similarly, the parametric equation of
and we know that
We now look into the graphs of these hyperbolic functions. Right now, just assume the derivatives I have used. A full description of them is given in the next section. For
we conclude the graph crosses the y-axis and has a horizontal tangent at the point (0,1). Looking at the second derivative we have
which tells us that the graph concaves up everywhere and so takes the form below.
A point to note is that by using the given definitions, we can sketch the graph Now let’s look at
The identity
By calculating the second derivative,
and check the range to find
We conclude that the graph is concave up for
Upon further inspection, we notice that the graph
and their difference Next up, we will look at the derivatives and integrals of these hyperbolic functions. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and
and 
and 
will lie on unit circle of the equation 
because
lies on the right branch of the curve given by 
because of the identity
. Since this curve is called a hyperbola, we thus accordingly label the functions sinh and cosh as hyperbolic functions. A quick fact here, though we will not prove, is that t is twice the area of the shaded hyperbolic section in the figure below.
, we notice the fact that 
and conclude that the graph is symmetric about the y-axis. Since
by geometrically adding the two curves 
and 
.
. This graph passes through the origin since
tells us that the graph is symmetric about the origin. We also know that the graph is rising at every point because
and concave down for 
and the point (0,0) is the only point of inflection.
lies above the graph 
because
as 
.
