Why cosh(1) - cos(1) Is Close to 1 but Not Exactly 1

The expression cosh(1) - cos(1) might appear puzzling at first. Upon closer inspection, we find that although it is not exactly 1, it is very close to 1, especially when using certain approximations. This phenomenon can be clarified through the Taylor series expansion of both cos(x) and cosh(x).

Introduction to Taylor Series Approximation

The Taylor series expansion is a powerful mathematical tool to approximate functions as polynomials. For cos(x) and cosh(x), we can use the following approximations:

cos(x) Approximation: [cos(x) approx 1 - frac{x^2}{2} frac{x^4}{24}] cosh(x) Approximation: [cosh(x) approx 1 frac{x^2}{2} frac{x^4}{24}]

Notice the striking similarity in the expressions. By subtracting the cosine approximation from the hyperbolic cosine approximation, we get:

[cosh(x) - cos(x) approx x^2]

This approximation is particularly useful for small values of x, but as x increases, the accuracy diminishes.

Calculating cosh(1) - cos(1)

When x 1, we can calculate the values of cosh(1) and cos(1) using their Taylor series approximations and compare their difference.

The Taylor series expansions for small angles in radians are:

cos(1) (1 radian): [cos(1) approx 1 - frac{1^2}{2} frac{1^4}{24} 1 - frac{1}{2} frac{1}{24}] cosh(1): [cosh(1) approx 1 frac{1^2}{2} frac{1^4}{24} 1 frac{1}{2} frac{1}{24}]

Following these approximations, we can calculate:

[cosh(1) - cos(1) approx left(1 frac{1}{2} frac{1}{24}right) - left(1 - frac{1}{2} frac{1}{24}right) 1.00277778 - 0.54030230586 0.54030236586 - 0.54030230586 0.47253995705]

However, the difference is not quite 1, but it is close. For more precise calculations, we can use numerical methods to find:

[cosh(1) 1.543080635]

[cos(1) 0.54030230586]

[cosh(1) - cos(1) 1.002778329]

Similarly, for 1 degree, we have:

[cosh(1^circ) 1.000152275]

[cos(1^circ) 0.99984769516]

[cosh(1^circ) - cos(1^circ) 0.00030457984]

As we can see, the difference is extremely small when dealing with angles in degrees, confirming the close approximation.

Conclusion

The approximation of cosh(x) - cos(x) approx x^2 is a useful tool for understanding the behavior of these functions near x 1. While it is not exact, it provides a good estimate for small values of x. This difference not being exactly 1 is a fascinating property of these trigonometric and hyperbolic functions.

Keywords: hyperbolic cosine, cosine, Taylor series approximation