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Hyperbolic Functions

Expository Paper

Masters Thesis

Amy L. Schutz

In partial fulfillment of the requirements for the Master of Arts in Teaching with a

Specialization in the Teaching of Middle Level Mathematics

in the Department of Mathematics.

Jim Lewis, Advisor Livia Miller, Advisor

July 2008



Introduction

This is a paper about hyperbolic functions; a class of functions that are, as we

shall see, analogous to the trigonometric functions. We will discuss the history of the

development of hyperbolic functions, their properties, their relationship to trigonometric

functions and to hyperbolas, and explore some applications of these interesting functions.

Hyperbolic functions arise quite naturally from simple combinations of the

exponential function, e x, a function that is explored thoroughly in school mathematics.

Indeed, the two main functions, the hyperbolic cosine and the hyperbolic sine are just half

of the sum and difference of e x

and e-x

, namely:

2)sinh(

2)cosh(

x x

x x

ee x

ee x

Of course, while it is natural to create these combinations of the exponential function, our

choice of names for these two functions raises the question of where the names come

from. We intend to answer this question and more.

History

An Italian mathematician, Vincenzo Riccati, developed the hyperbolic functions

in the mid-18th century. Ricatti studied hyperbolic functions and employed them to obtain

solutions of cubic equations. He also found the standard addition formulas, similar to

trigonometric identities, for hyperbolic functions as well as their derivatives. Ricatti’s

approach did not begin with the definition we gave above. Instead, with some work done



jointly with Saladini, he found the relationship between the hyperbolic functions and the

exponential function. His work was published between 1757 and 1767. Ricatti used Sh.

and Ch. as notation for the hyperbolic sine and cosine and used the similar abbreviations

of Sc and Cc for the circular functions, sine and cosine.

Despite what is now a general agreement that Ricatti first introduced the

hyperbolic functions, a French mathematician by the name of Johann Heinrich Lambert

has often been credited for their introduction. In 1768, Lambert published his work

concerning on hyperbolic functions, in which he made the first systematic development

of their theory. Thus Lambert became associated with hyperbolic functions in the same

way that Leonard Euler is associated with circular functions; Lambert popularized the

new hyperbolic trigonometry as it is used in modern science. Lambert began using the

notation of sinh x, the Latin sinus hyperbolus, and cosh x to represent these functions.

This notation is widely used today although in some cases one can also find the notation

hysin and hycos that George Minchin introduced in 1902.

Although hyperbolic functions were not introduced until around the late 1750’s,

almost two hundred years earlier Gerhard Kremer made the first and one of the most

important applications of the functions we now know as hyperbolic. Kremer was a

Flemish geographer who was better known by his Latin name, Gerhardus Mercator. In

1569, he issued his Mercator’s Projection map. The projection resulted in the making of a

map in which a straight line always made an equal angle with every meridian. This was a

major breakthrough in navigation and is used today in all deep-sea navigation charts of

the world. However, when he published his work, it was left to others to discover that the

formulas he used were linked to the hyperbolic functions.



Modern interest in hyperbolic functions increased with the invention and

commercialization of electricity. The widespread use of electricity started in the mid

1850’s with the development of telegraphs. This was followed by the telephone, electric

light, and the need of industry for power. Generating plants and transmission lines started

to cross America and this transmission of electrical power involves the application of

hyperbolic functions. Finally, during the early decades of the 1900’s, applications of

hyperbolic functions could be found in every scientific discipline.

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric or circular functions. Just as the

trigonometric functions are related to the circle, the hyperbolic trigonometric functions

are related to the standard hyperbola: 122 y x for x > 1. For example, using our

definition above, we can show that .1)(sinh)(cosh 22 x x Namely,

22

22

22sinhcosh

x x x x

eeee x x

44

22 x x x x eeee

22

4

1 x x x x eeee

x x x x x x x x

eeeeeeee

2222

224

1

x x x x x x x x eeeeeeee 2222 224

1

x xee44

1



x xee

1

We can also use circular trigonometric identities, and some information about complex

numbers, to help us derive the formulas for the hyperbolic functions.

There exist the following circular trigonometric identities:

)sin()sin( x x

)cos()cos( x x

ixe xi x sincos

If we take the identity ixe xi x sincos and replace x in this identity with – x, we get the

following,

)()sin()cos( xie xi x

Since )cos()cos( x x and )sin()sin( x x , we can substitute these values into this

equation and

)(

)sin()cos(xi

e xi x

now becomes

ixe xi x )sin()cos( .

If we take the last equations and add it to the original identity, we get the following:

ix

ix

ix

e

e xi x

e xi x

ixe2cos(x)

)sin()cos(

)sin()cos(

We can now divide both sides by two and get

2)cos(

ixix ee x .



Similarly, we could have subtracted the two equations to obtain the following:

ix

ix

ix

e

e xi x

e xi x

ix

e2isin(x)

))sin()cos((

)sin()cos(

Again, we can divide both sides by two and get

i

ee x

ixix

2)sin(

The two equations for sine and cosine,2

)cos(ixix ee

x andi

ee x

ixix

2)sin( , are

called Euler’s equations.

Since the identity ixe xi x sincos is valid for complex as well as real numbers

x, we can replace each x in the Euler’s equations with ix to obtain

2cos

x x eeix

and

2sin

x x eeiix .

This provides a different justification for using the exponential expressions in the right-

hand members of these two equations to define the hyperbolic cosine and sine functions,

cosh and sinh. Thus,

2cosh:),(cosh

x x ee

x y y x

and

2sinh:),(sinh

x x ee x y y x



We build on this connection to trigonometric functions to define the hyperbolic tangent

of x, denoted tanh( x), as follows:

x x

x x

x x

x x

eeee

ee

ee

x x x

)(2

1

)(

2

1

)cosh()sinh()tanh( .

(Beckenback, Dolciani & Wooton, 1966)

It must also be noted that in 1759 Daviet de Foncenex showed how to interchange

circular and hyperbolic functions using the number i= 1 . Also, other mathematicians

contributed to the foundations of this work. For example, DeMoivre’s classical equation

of 1722, )sin(cos)sin(cos nxinx xi x n underpinned Daviet de Foncenex’s findings.

Euler’s Equation followed this in 1748, )sin(cos xi xe ix . As a result, we have

familiar links between the circular and hyperbolic functions of:

iii coshcosand,sinhsin

and the converse equations:

iii coscoshand,sinsinh

The use of these identities leads us to many substitutions that can readily be made

between the circular and hyperbolic functions.

We find that the theory of hyperbolic functions is analogous to the theory of

circular functions. As expected, the hyperbolic functions can also be compared to the

circular trigonometric functions graphically. To demonstrate this we first discuss the

theory of circular functions and follow that with the theory of hyperbolic functions. The

graphs that follow were constructed using the software GeoGebra and much of the



exposition is based on a discussion of hyperbolic functions that can be found in

Shervatov’s Hyperbolic Functions.

Theory of Circular Functions

Consider the unit circle 122 y x . Let B be an arbitrary point of the unit circle

in the x-y coordinate plane. If point B is a point of the unit circle, then the measure of

angle BOA (in radians), call it , between the radii OA and OB is defined to be the

length of the arc AB (denoted AB) or, alternatively, twice the area of the sector AOB

determined by the arc AB and the radii OA and OB. (Note: A fundamental property of

the angle, , of the circle is that is not changed by a rigid rotation of the sector AOB

about the origin.)



Draw segment BC perpendicular to OA. Also, draw a line tangent to the unit

circle at point A; let D be the point of intersection of this tangent with the line defined by

extending OB.

The lengths of the segments BC ,OC and AD (denoted BC, OC, and AD) are

equal to the sine, cosine, and tangent of the angle BOA, respectively:

BC = sin , OC = cos , AD = tan

Theory of Hyperbolic Functions

Consider the unit hyperbola 122 y x . Let A be an arbitrary point on the unit

hyperbola, then the measure of the hyperbolic angle, t , between the radii OB and OA is

defined to be twice the area of the sector BOA determined by the arc AB of the hyperbola

and the radii OB and OA.



Note that hyperbolic angle in standard position is the angle at (0,0) between the

ray to (1,1) and the ray to ( x, 1/ x) where x > 1. In other words, hyperbolic angles are

formed by the intersection of hyperbolic rays analogous to the formation of angles in

Euclidean Geometry. The measure of a hyperbolic angle, BAC is defined to be the

measure of the Euclidean angle, B'AC', formed by the Euclidean tangent lines, AB' and

AC'. Also, a fundamental property of the hyperbolic angle t of the hyperbola is that t is

not changed by a hyperbolic rotation of the sector BOA.

Draw the perpendicular AC from point A, on the unit hyperbola, to OB. Note

that the line defined by OB is an axis of symmetry of the hyperbola, and the point B is a

vertex of the hyperbola. Also, draw a line tangent to the hyperbola at point B; let D be the

point of intersection of this tangent with OA.

The lengths of the segments AC , OC , and BD are called, respectively, the

hyperbolic sine, the hyperbolic cosine, and the hyperbolic tangent of the hyperbolic angle

t ; we write AC = sinh t , OC = cosh t , and BD = tanh t .

There are deep similarities between the circular and the hyperbolic functions.

Note that reciprocal functions of sinh(x), cosh(x), and tanh(x) exist just as they do for

circular functions. Predictably, they are called csch(x), sech(x), and coth(x) respectively.

However there are also differences. For example, we know that circular functions vary in

Graph is from NonEuclid web-site:

http://www.beva.org/math323/asgn6/exercise.htm#Angles



a periodic fashion: the sine and cosine functions each have period 2 , and the tangent

function has period . On the other hand, the hyperbolic functions are not periodic.

Graphs

We can see some of the differences between circular and hyperbolic functions in

their graphs (this would be expected since circular functions are periodic and hyperbolic

functions are not). First we can compare circular sine curve to the hyperbolic sine curve:

Comparing the sin(x) and sinh(x) graphs, we can see that the ranges for these

functions are the following:

Range of sin(x) = 11 x x

Range of sinh(x) = x x

We note that the intersection of the two functions is the coordinate point (0,0).



We can also compare cosine and hyperbolic cosine:

The circular cosine function has the same range as the circular sine function.

However, the ranges for the hyperbolic sin and cosine are different. The ranges are now

the following:

Range of cos(x) = 11 x x

Range of cosh(x) = x x 1

The intersection point of the circular cosine and hyperbolic cosine functions is the

coordinate point (0, 1). This graph also shows the periodicity of the circular cosine

function.



Finally, we can compare tan(x) and tanh(x):

Again, we can see the ranges between the circular tangent and hyperbolic tangent

functions are different. They are the following:

Range of tan(x) = x x

Range of tanh(x) = 11 x x

We can also use graphs to examine the connections between hyperbolic and

exponential functions. Recall that the definition of the hyperbolic functions used the

exponential function, x

e . The graphs of the sum and difference of sinh(x) and cosh(x)

and their connection to the graphs of x

e and xe can be displayed graphically as follows:



The graph gives us a visual sense that the graphs of the exponentials are actually the sum

and differences of the two hyperbolic functions. This supports the following identities

that we will be verified below.

)sinh()cosh( x xe x

and

)sinh()cosh( x xe x

Identities

Several circular trigonometric identities exist. There also exist several hyperbolic

identities similar to these circular identities. We will develop some of the identities along

with list a few of the other identities that exist.

We will begin with the proof of the identity: )sinh()cosh( x xe x . The

definitions for hyperbolic sine and cosine have already been established. If we substitute



the definitions into the identity and simplify we can see that the sum of the two

hyperbolic functions is equivalent tox

e . Here is the argument:

22)sinh()cosh(

x x x x eeee x x

2

x x x x eeee

x x

ee

2

2

We can follow the same procedure for the identity with the difference of the two

hyperbolic functions, )sinh()cosh( x xe x . Here is the argument:

22)sinh()cosh(

x x x x eeee x x

2

)( x x x x eeee

2

x x x x eeee

x x

ee

2

2

The proofs of the identities support the graph in the previous section.

Other identities of circular functions also correspond to identities of hyperbolic

functions. We may know some of them as a definition,cos

sintan . Similarly

)cosh(

)sinh()tanh(

t

t t . If we examine the similar triangles found in the first diagrams of the

hyperbolic functions section separately, we can see the depth of the similarities between



the identities in circular and hyperbolic functions. First we can look at the similar

triangles in the circular function.

The two triangles OCB and OAD are similar to each other. It then follows that

AD

OA=

BC

OC .

However, we can embed these triangles in the unit circle, making OA the radius and thus

of length one. This gives us

AD=

BC

OC

But we previously have stated that BC = sin , OC = cos , and AD = tan so we then

obtain

cos

sintan .

Analogous to this circular trigonometric identity, there exist the hyperbolic

function identity,)cosh()sinh()tanh(t t t . Again, if we look at the similar triangles in the first

two graphs of the hyperbolic functions we get the following similar triangles:



From the similar triangles OBD and OCA, it then follows that

OC CA

OB BD

However, (embedding the graph onto the unit hyperbola), OB equals one. This gives us

OC

CA BD

But we previously have stated that CA = sinh(t ), OC = cosh(t ), and BD = tanh(t ) so we

then obtain

)cosh(

)sinh()tanh(

t

t t .

We can also use these diagrams to help us with a Pythagorean Identity. We are

aware that the Pythagorean Theorem is stated as 222 cba where a and b are the legs

of a right triangle and c is the hypotenuse.

Focusing on the analogous hyperbolic function and its graph yields the following

illustration and discussion.



The coordinate of the point A on the hyperbola is (OC, CA), where OC is the x-

coordinate and CA is the y-coordinate. The equation for the unit hyperbola can be written

in the form x2- y

2=1. We can substitute x and y with their equivalent values and it

follows that 122 CAOC . Again, we have already stated that CA = sinh(t ) and OC =

cosh(t ). Substituting again we get the identity for hyperbolic functions that

1sinhcosh 22 t t

We have previously used our analytic definition of cosh (x) and sinh (x) to establish this

identity. Two more identities can easily be derived from this identity. If we divide both

sides by t 2cosh and simplify we get

t t

t t 22

22

cosh

1

cosh

sinhcosh

or

t t

2

2

cosh

1tanh1 .



Following the same procedure but dividing by t 2sinh results in the following identity

t t

2

2

sinh

11coth .

We can also convert from a circular function identity to the analogous hyperbolic

identity by using Osborne’s Rule. Osborne’s Rule states that any circular identity can be

converted to an analogous hyperbolic identity by expanding, exchanging circular

functions with their corresponding hyperbolic functions, and then flipping the sign of

each term that involves the product of two sinh( x). Two examples can be used to show

the use of this rule.

The first example starts with the circular identity 1cossin 22 x x . If we expand

the identity we get

1coscossinsin x x x x .

Next, Osborne’s Rule says to exchange the circular functions with their corresponding

hyperbolic functions and you get

sinh xsinh x+cosh xcosh x =1.

Since we now how an equation that involves the product of two hyperbolic sines, we can

then flip the sign of this term. Following Osborne’s Rule we end up with

1coshcoshsinhsinh x x x x

By simplifying the products into powers and switching the order of the terms we get the

identity that was proven earlier

1sinhcosh 22 x x .

The second identity using Osborne’s Rule requires fewer steps due to the format of the

identity. First we take the trigonometric identity



y x y x y x sinsincoscos)cos(

The identity is already expanded and we can begin by substituting in the hyperbolic

functions to give us

y x y x y x sinhsinhcoshcosh)cosh(

Now we can change the sign of the term that involves the product of two hyperbolic sines

and get the analogous hyperbolic identity of

y x y x y x sinhsinhcoshcosh)cosh( .

The hyperbolic sine function is an odd function, which is,

11 sinh)sinh( x x . Therefore, the hyperbolic sine function is symmetric with respect

to the origin. Again, this is analogous to the circular function where sin(-x)=-sinx. To

prove this we will use the definition of sinh(x) and replace x with – x.

)(2

1sinh x x ee x Original sinh(x) equation

)(

2

1)sinh( )( x x ee x Substitute (-x) for x

)(2

1)sinh( x x ee x Simplify exponents

)(2

1)sinh( x x ee x Factor out a negative

)(2

1)sinh( x x ee x Rearrange terms in parenthesis

)sinh()sinh( x x Substitute

We can also prove that the hyperbolic cosine function is an even function, that is

11 xcosh)cosh( x . Therefore, the hyperbolic cosine function is symmetric with respect



to the y-axis. Again, this is analogous to the circular function where cos(-x)=cosx. We

will use the definition of cosh(x) to prove this.

)(

2

1cosh x x ee x Original cosh(x) equation

)(2

1)cosh( )( x x ee x Substitute (-x) for x

)(2

1)cosh( x x ee x Simplify exponents

)(2

1)cosh( x x ee x Rearrange terms in parenthesis

)cosh()cosh( x x Substitute

Several other identities exist for both circular and hyperbolic functions that are

analogous to each other. By using the definition of hyperbolic funct ions and Osborne’s

Rule, you can verify these identities and their similarities.

Catenaries

Galileo believed that the shape of a chain, rope or cable that is suspended at two

ends and under the influence of gravity was a parabola. However, the equation for the

shape is y = cosh(x) while a parabola is y = x2. Christiaan Huygens named the shape a

“catenary” after the Latin word “catena”, meaning chain. It is easy to see where Galileo

would think that the shape was a parabola. In fact, a parabola and a catenary are very

similar near the vertex. However, as the absolute value of x increases, the catenary

rapidly outgrows the parabola. This can be seen in the graph of a parabola and a catenary

below.



Even though a parabola is not the same as a catenary, they are related. If a

parabola is moved along a straight line, the path of the focus of the parabola is a catenary.

It is also known that if a bicycle contained polygonal wheels, for example a square, it

would travel smoothly on a road made of inverted catenaries and the rider would stay at

the same height. A visual of this can be seen on the by following the link:

http://mathworld.wolfram.com/Roulette.html. A final characteristic of catenaries is that

their center of gravity is lower than that of any curve of equal perimeter with the same

fixed points.

A real-life example of an upturned catenary is the St. Louis Arch. Catenaries can

also be seen on turbine wheels. Turbine wheels consist of a set of blades extending

inward toward the axis. The blades are mounted at their outer ends on the inside of the

rim and the inner ends of the blades are attached on the outside of the axial hub. Each

http://mathworld.wolfram.com/Roulette.html





blade on the turbine wheel has its longitudinal tensile elements shaped in the form of part

of a relaxed catenary with a bow in the direction of force of fluid current on the blade to

provide a practical wheel of reasonable cost. The relaxed catenary configuration may also

be seen in struts and strut vanes supporting hub elements within the nozzle. In addition to

the use of catenaries in the arch and in the turbine wheels, catenaries are important in the

overhead wires on railroad electrifications. Trolleys depend on these pantographs with

catenary formation to help prevent them from bouncing off of the wires.

Applications

We have seen a few examples of the uses of catenaries. It is important to note that

hyperbolic functions have other uses besides the uses of catenaries. For example, the

hyperbolic tangent function can be used to calculate the speed of ocean waves. In

addition to this, the hyperbolic cosine and its inverse are encountered in fluid friction in

the case of quadratic drag. The wave speed is calculated using the following formula:

d g v 2tanh

2

where is the wavelength, d is the depth, and g is the acceleration of gravity.

Conclusion

We have found that the hyperbolic functions are analogous to the trigonometric

functions. We have also shown how they can be derived from simple combinations of the

exponential functions. This deep relationship to familiar functions, allowed us to uncover



many properties that we illustrated by focusing on two of the main hyperbolic functions:

the hyperbolic sine and hyperbolic cosine. Recall that we defined these as

2)sinh(

2)cosh(

x x

x x

ee x

ee x

We have discussed the development, properties, and identities of hyperbolic

functions including the contributions by Vincenzo Riccati and Johann Heinrich Lambert.

Because of the work done by these mathematicians, and others, we are able to use

hyperbolic functions in every scientific discipline. We only focused on a few uses;

catenaries are used in power lines and trolleys and hyperbolic tangent is used in

calculating the speed of ocean waves. However, there are many more including several in

the field of physics and throughout the sciences. This is clearly an intriguing subject that

lends itself to continued study.



References

Baker, C. C. T.(1961). Dictionary of Mathematics. Hart Publishing Company, Inc.

Calculus and Analytic Geometry (Alternate Edition) (1972). Thoma, Beorge B. Jr.

Addison Wesley

Modern Trigonometry (Teacher’s Edition) (1966). Beckenback, Edwin F., Dolciani,

Mary P., Wooton William, Houghton Mifflin Company

Shervatov, V.G.(1963). Hyperbolic Functions. The University of Chicago

Thomas’ Calculus Eleventh Edition (2005). Giordano, Frank R., Hass, Joel, Weir,

Maurice D., Pearson Addison Wesley

http://faculty.matcmadison.edu/kmirus/Reference/HyperbolicFunctions.doc

http://math.furman.edu/~dcs/book/c6pdf/sec67.pdf

http://www.sosmath.com/trig/hyper/hyper01/hyper01.html

http://www.math10.com/en/algebra/hyperbolic-functions/hyperbolic-functions.html

http://www.webgraphing.com/examples_transcendentals.jsp?goog=hyperbolic_trig

&gclid=CLWu4Kfc4ZQCFRKpxgodGiT3SA

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http://planetmath.org/encyclopedia/HyperbolicFunctions.html

http://www.economicexpert.com/a/Catenary.htm

http://www.economicexpert.com/a/Hyperbolic:function.html

http://www.faculty.fairfield.edu/jmac/sj/scientists/riccati.htm

http://www-groups.dcs.st-andrews.ac.uk/~history/Curves/Catenary.html

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http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page5.htm



http://demonstrations.wolfram.com/RouletteAComfortableRideOnAnNGonBicycle/

http://planetmath.org/encyclopedia/HyperbolicAngle.html


http://hyperphysics.phy-astr.gsu.edu/hbase/watwav.html#c3






















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