If you work in engineering or physics, you have to work with a lot of models. Well, to solve modeling problems you'll need hyperbolic functions.
Certain even and odd combinations of the exponential functions ex and e-x arise so frequently in engineering, physics, and mathematics that they're given special names. Hyperbolic functions are analogous trigonometric functions in that they are named the same as trigonometric functions with the letter 'h' appended to each name.
These special functions have the same relationship to the hyperbola that trigonometric functions have to the circle. For this reason, they are collectively known as hyperbolic functions and are individually called hyperbolic sine, hyperbolic cosine, and so on. In addition to modeling, they can be used as solutions to some types of partial differential equations.
Think of taking a rope, fixing the two ends, and then letting it hang under the force of gravity. The shape of the rope will naturally form a hyperbolic cosine curve.
Types & Graphs
Hyperbolic Sine: y = sinh(x)
This math statement is read as 'y equals hyperbolic sine x' or 'sinch x.'
The exponential formula is:
Note that the domain and range of this function are both the set of real numbers. This function is also one-to-one.
Hyperbolic Cosine: y = cosh(x)
This math statement is read as 'y equals hyperbolic cosine x' or 'cosh x.'
Note that the graph is symmetric with respect to the y-axis and it has a minimum vertex at (0, 1). The domain of this function is the set of real numbers and the range is any number equal to or greater than one.
Hyperbolic Tangent: y = tanh(x)
This math statement is read as 'y equals hyperbolic tangent x' or 'tanch x.'
Note that the graph is symmetric with respect to the origin. The lines y = 1 and y = -1 are both horizontal asymptotes. The domain of the function is the set of real numbers. The range is the open interval (-1, 1).
Hyperbolic Cotangent: y = coth(x)
This math statement is read as 'y equals hyperbolic cotangent x' or 'coth x.'
Note that the graph is symmetric with respect to the origin. The lines y = 1 and y = -1 are both horizontal asymptotes. The y-axis is a vertical asymptote. The domain of the function is the set of real numbers excluding zero. The range is (-infinity, -1) union (1, infinity).
Hyperbolic Secant: y = sech(x)
This math statement is read as 'y equals hyperbolic secant x.'
The x-axis is a horizontal asymptote. The graph is symmetric with respect to the y-axis. The domain of the function is the set of real numbers. The range of the function is the half-open interval (0, 1].
Hyperbolic Cosecant: y = csch(x)
This math statement is read as 'y equals hyperbolic cosecant x.'
Note that both the x- and y-axes are asymptotes. The graph is symmetric with respect to the origin. Both the domain and range of the function are the set of real numbers excluding zero.
If t is any real number, then the point P(cosh t, sinh t) lies on the right branch of the hyperbola x2 - y2 = 1 since cosh2t - sinh2t = 1 and cosht > = 1. It turns out that t represents twice the area of the shaded hyperbolic sector in the figure:
A practical application of cosh and sinh is a hunter pursuing his prey while riding on a stallion. The hunter's weapon of choice are bolas, a rope with weights at two ends. When the hunter hurls the bolas at the prey, the weights at the end of the taught rope must pack enough energy to tangle around the legs and hold.
The equation of motion of the weights is given by:
where x is the position of the weights at any given time and omega is the rotational velocity of the weights. The second derivative of x with respect to t (the left hand side of the equation) is the linear acceleration of the weights. This equation can be solved for the velocity of the weights. One can show the impact velocity of the weights hitting the poor prey is:
Ouch! That would hurt!
Another application occurs when we look at ocean waves. The velocity of a water wave with length L moving across a body of water with depth d and gravitational acceleration g is modeled by the function:
So the next time you're surfing on the North Shore, bring your calculator so that you can compute the water depth needed to catch the big wave.
The List of Properties
A famous hyperbolic identity is:
The derivatives of cosh and sinh are each other:
Those two functions also have series expansions:
Hyperbolic functions are a class of functions that are used to solve problems arising in oceanography, engineering, physics, and math. They are linear combinations of ex and e-x and are analogous to their circular function counterparts.
In circles, for any real number t, cos2t + sin2t = 1 and the point (cost, sint) lies on the unit circle. Hence, sin(t) and cos(t) are called circular functions. Similarly for real t, cosh2t - sinh2t = 1. In this case, each point (cosht, sinht) lies on the hyperbola x2 - y2 = 1 making sinh(t) and cosh(t) hyperbolic functions.
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Hyperbolic Trigonometric Functions: Practice Problems
- Hyperbolic Trigonometric Functions: Functions related to Hyperbolas
Practice Problems (Show all your work)
(a) Find all six hyperbolic trigonometric function values for the angle 60 degrees
Answers (To Check Your Work) -
(a) First, we need to convert our angle from degrees to radians as follows:
x = 60 degrees = 60 x pi/180 = pi/3 radians.
Next, we use the radian measure and a calculator to calculate the six hyperbolic trigonometric function values for our angle of x = pi/3 radians:
- sinh (pi/3) = (e(pi/3) - e(-pi/3)) / 2 = 1.25.
- cosh (pi/3) = (e(pi/3) + e(-pi/3)) / 2 = 1.60.
- tanh (pi/3) = (e(pi/3) - e(-pi/3)) / (e(pi/3) + e(-pi/3)) = 0.78.
- coth (pi/3) = (e(pi/3) + e(-pi/3)) / (e(pi/3) - e(-pi/3)) = 1.28.
- csch (pi/3) = 2/ (e(pi/3) - e(-pi/3)) = 0.80.
- sech (pi/3) = 2/(e(pi/3) + e(-pi/3)) = 0.63.
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