Hyperbolic Functions & Addition Formulas: Calculations & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we define two basic hyperbolic functions and then extend these ideas to the other hyperbolic trig functions. Also, the hyperbolic sine and hyperbolic cosine of the sum of two numbers is defined, verified and used in examples.

Hyperbolic Functions

Sometimes a concept can appear extremely abstract and yet be really easy. For example, the hypotenuse of a right triangle is just the longest side. And, a hypothesis is an educated guess.

To avoid getting hyper as we hyper-extend our hyper vocabulary, we'll proceed one step at a time to explore the idea of hyperbolic functions. These functions pop up all the time in engineering and science applications involving waves, so let's learn more!

In this lesson we define the two basic hyperbolic functions and then show how these two can lead to four other hyperbolic functions. Also, we will look at the hyperbolic function of the sum of two numbers.

Some Basic Hyperbolic Functions

Sinh

The hyperbolic sine is sinh (pronounced 'sinch'), We can evaluate this function using:


sinh(a)=(e^a-e^-a)/2


For example, the sinh(2):


sinh(2)=(e^2-e^-2)/2=3.6269


Likewise,


sinh(1)=(e^1-e^-1)/2=1.5431


Cosh

Just like sine and cosine, we have sinh and cosh. The definition for cosh:


cosh(a)=(e^a+e^-a)/2


Do you agree with the following?


cosh(2)=3.7622


and


cosh(1)=1.5431


Tanh

Remember how tangent is sine over cosine? Well, the hyperbolic tangent, tanh (pronounce ''tanch'') is:


tanh(a)=sinh(a)/cosh(a)


And this idea continues for the other hyperbolic functions:

Cotangent, Sech and Csch

The hyperbolic cotangent:


coth(a)=1/tanh(a)=cosh(a)/sinh(a)


The hyperbolic secant, sech:


sech(a)=1/cosh(a)=2/(e^a+e^-a)


And the hyperbolic cosecant, csch:


csch(a)=1/sinh(a)=2/(e^a-e^-a)


Hyperbolic Functions of a + b

Just like the sine of the sum of two numbers, we have a hyperbolic sine of the sum of two numbers:


sinh(a+b)=sinh(a)cosh(b)+cosh(a)sinh(b)


There's one for the cosh as well:


cosh(a+b)=cosh(a)cosh(b)+sinh(a)sinh(b)


What if we wanted to verify these results using the basic definition of sinh and cosh?

Let's look at how we could do this for cosh(a + b). Let's start with the right-hand side (RHS):


RHS_of_cosh(a+b)


Now, we substitute those exponential definitions for sinh and cosh:


RHS_of_cosh(a+b)_written_in_terms_of exponentials


Expanding the products:


expanding_the_products


See how the ''blue'' terms will cancel? Same thing happens with the ''green'' terms. This leaves us with:


cancelling_terms


The 2's cancel with the 4 leaving a 2 in denominator. Also, -a - b is written as -(a + b):


RHS=(e^(a+b)+e^(-a-b))/2


What if we substitute u for a + b:


RHS=(e^u+e^-u)/2


But this is how we define cosh. Thus,


RHS=cosh(u)


And, u is a + b:


RHS=cosh(a+b)


which is the same as the left-hand side of the cosh(a + b) formula. Thus, we have verified the cosh(a + b) expression. If we wanted to, we could use this same method to verify the sinh(a + b) expression.

Some Practice With Numbers

Let's test these results with a numerical example by calculating sinh(3) two ways. Using our exponential definition of sinh and a calculator we get:


sinh(3)=10.0179_approx_10.018


For convenience, the results from the calculator are written as numbers with four decimal places. After we do the product and sum calculations, we will approximate the result with only 3 decimal places to allow for round-off errors.

So, the final answer for sinh(3) is 10.018.

What if we think of 3 as being the sum of 2 and 1. Then, we can use the sinh(a + b) formula:


sinh(2+1)=10.018_using_the_formula


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