Relationships between hyperbolic and trigonometric functions

Hyperbolic functions are analogous to traditional trigonometric functions, but for a hyperbola rather than a circle. While trigonometric functions such as sine and cosine are defined based on a unit circle, hyperbolic functions are based on a unit hyperbola. They have many applications in mathematics, physics, and engineering, often providing solutions to problems that cannot be easily solved by trigonometric functions alone.

Understanding hyperbolic functions

Like the trigonometric functions, the hyperbolic functions also have identities and properties that relate them to each other and to the exponential functions. The elementary hyperbolic functions are the hyperbolic sine and cosine functions, denoted as sinh and cosh.

Hyperbolic sine and cosine are defined as follows:

sinh(x) = ( ex - e -x ) / 2
cosh(x) = (e x + e -x ) / 2

From these we can obtain other hyperbolic functions:

tanh(x) = sinh(x) / cosh(x)
coth(x) = cosh(x) / sinh(x)
sech(x) = 1 / cosh(x)
csh(x) = 1 / sinh(x)

Visualizing a hyperbolic function

Hyperbolic sine (sinh) and cosine (cosh)

The graphs show how sinh(x) and cosh(x) behave similar to sine and cosine, with smooth curves emanating from the origin. However, these functions grow rapidly as their inputs increase.

Viewing other hyperbolic functions

Above, the functions tanh(x) and coth(x) are shown. Note that tanh(x) approaches +1 or -1 as x approaches infinity or negative infinity, while coth(x) starts at zero and increases.

Relation to trigonometric functions

Although they model different geometric concepts (circle vs. hyperbola), there is an interesting connection between the hyperbolic and trigonometric functions. This occurs through the complex numbers, where traditional trigonometric identities and formulas have hyperbolic equivalents.

Exponential relationship

The key is Euler's formulas, which relate complex exponentials to trigonometric functions:

e ix = cos(x) + i*sin(x)
e -ix = cos(x) - i*sin(x)

You'll find that the hyperbolic functions are similar but don't involve complex numbers directly:

e x = cosh(x) + sinh(x)
e - x = cosh(x) - sinh(x)

Conversion between hyperbolic and trigonometric functions

There are several known transformations between trigonometric and hyperbolic functions via complex numbers:

• sinh(x) = -i * sin(ix)
• cosh(x) = cos(ix)
• tanh(x) = -i * tan(ix)

Examples and exercises

Example 1

Find the value of sinh(ln(3)).

Use of the definition:
sinh(x) = ( ex - e -x ) / 2

Let x = ln(3):

sinh(ln(3)) = (e ln(3) - e -ln(3) ) / 2
            = (3 - 1/3) / 2
            = (9/3 - 1/3) / 2
            = 8/6
            = 4/3

Example 2

Compute cosh(0) and interpret it geometrically.

Use of the definition:
cosh(x) = (e x + e -x ) / 2 

At x = 0:
cosh(0) = (e 0 + e -0 ) / 2
         = (1 + 1) / 2
         = 1

Explanation:
This result shows that the value of the hyperbolic cosine function is the same as the cosine at 0 on the unit circle, showing its similarity with the geometry of the circle.

Exercise 1

Prove that sinh^2(x) - cosh^2(x) = -1.

evidence:
sinh(x) = ( ex - e -x ) / 2
cosh(x) = (e x + e -x ) / 2

Calculate sinh^2(x):
sinh^2(x) = [(e x - e -x ) / 2] 2
          = (e 2x - 2 + e -2x )/4

Calculate cosh^2(x):
cosh^2(x) = [(e x + e -x ) / 2] 2
          = (e 2x + 2 + e -2x )/4

Now subtract:
sinh^2(x) - cosh^2(x) 
= [(e 2x - 2 + e -2x ) - (e 2x + 2 + e -2x )] / 4 
= -4/4 
= -1

Applications and insights

Hyperbolic functions are not just theoretical constructs; they also have practical applications, especially in the fields of engineering and physics. For example, they describe the shape of suspended cables, the motion of catenaries, and the properties of certain materials under tension.

These functions are also helpful in solving certain differential equations, especially those that describe phenomena associated with hyperbolic geometry. By understanding the relationship between hyperbolic and trigonometric functions, you gain a broader understanding of how various mathematical concepts are inherently connected.

For a deeper understanding, consider the formula connecting the hyperbolic tangent to exponential growth:

tanh(x) = (e 2x - 1) / (e 2x + 1)

This expression clearly shows the saturating behavior of tanh(x), which is similar to the potential limits in biological and technical systems.

Conclusion

The relationship between hyperbolic and trigonometric functions is a deep part of mathematics, revealing the interconnected nature of algebra, geometry, and calculus. By mastering these concepts, students gain powerful tools for analyzing a wide range of scientific and engineering problems.

Comments