Question

What is the fundamental identity for hyperbolic functions?

Short answer

Answer: The fundamental identity for hyperbolic functions is cosh^2(x) - sinh^2(x) = 1.

Step-by-step solution

Recall the definitions of hyperbolic sine and cosine functions

To find the fundamental identity for hyperbolic functions, we first need to recall the definitions of hyperbolic sine (sinh) and hyperbolic cosine (cosh) functions. They can be defined using the exponential function as follows: sinh(x) = \frac{e^x - e^{-x}}{2} cosh(x) = \frac{e^x + e^{-x}}{2}

Square both the hyperbolic sine and cosine functions

Now, we will find the squares of both the hyperbolic sine and cosine functions: sinh^2(x) = (\frac{e^x - e^{-x}}{2})^2 = \frac{e^{2x} - 2 + e^{-2x}}{4} cosh^2(x) = (\frac{e^x + e^{-x}}{2})^2 = \frac{e^{2x} + 2 + e^{-2x}}{4}

Calculate the difference between the squares of hyperbolic sine and cosine functions

Now, we will calculate the difference between the squares of the hyperbolic sine and cosine functions: cosh^2(x) - sinh^2(x) = \frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4}

Simplify the expression

We simplify the expression by combining the terms with the same exponent and canceling out the similar terms: cosh^2(x) - sinh^2(x) = \frac{e^{2x} + 2 + e^{-2x} - e^{2x} + 2 - e^{-2x}}{4} cosh^2(x) - sinh^2(x) = \frac{4}{4} Finally, we get the fundamental identity for hyperbolic functions: cosh^2(x) - sinh^2(x) = 1

Key concepts

Hyperbolic Sine

The hyperbolic sine function, often denoted as \(\sinh(x)\), plays a pivotal role in various areas of mathematics, including hyperbolic geometry, complex analysis, and the solutions of certain differential equations. You can think of \(\sinh(x)\) as a counterpart to the sine function found in trigonometry but for hyperbolic geometry.

The definition of hyperbolic sine using exponential functions is:
\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]
This represents the difference of two exponential terms divided by two. The function exhibits growth as \(x\) increases, similar to exponential functions, but with symmetry across the origin, showing that \(\sinh(-x) = -\sinh(x)\). Understanding this function is essential in deriving further identities and properties of other hyperbolic functions.

Hyperbolic Cosine

Complementary to hyperbolic sine is the hyperbolic cosine function, depicted as \(\cosh(x)\). If you're familiar with the cosine function from trigonometry, \(\cosh(x)\) is its hyperbolic analog, reflecting the properties of a cosine wave along a hyperbolic curve.

Defined by the expression:
\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]
This signifies the average of two exponential functions. Noteworthy is the even nature of \(\cosh(x)\), meaning that \(\cosh(-x) = \cosh(x)\). This evenness is part of the reason why the hyperbolic identity is so elegant and useful. Grasping both the hyperbolic sine and the hyperbolic cosine functions is crucial to fully understand the underlying hyperbolic geometry and to manipulate and simplify expressions involving hyperbolic functions.

Exponential Functions

Exponential functions are mathematical expressions that generally have a variable as the exponent. The basic form of an exponential function is \(f(x) = e^x\), where \(e\) is Euler's number, approximately equal to 2.71828. These functions are fundamental in calculus and appear in numerous applications such as compound interest, population growth, and radioactive decay.

They are distinct in their property of having a constant rate of growth, which is proportional to the function's current value. This self-relation leads to their ubiquity in modeling systems where growth accelerates over time. When dealing with hyperbolic functions, we rely on exponential functions to define them and establish their properties, as seen in the definitions of \(\sinh(x)\) and \(\cosh(x)\).

Identities in Calculus

Identities are equations that hold true for all values within the domain of the functions involved. They are the backbone of simplifications and problem-solving techniques in calculus. When dealing with hyperbolic functions, one of the most essential identities is the so-called fundamental identity, which relates the squares of hyperbolic sine and cosine.

The fundamental identity of hyperbolic functions is expressed as:
\[\cosh^2(x) - \sinh^2(x) = 1\]
This identity resembles the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\) from trigonometry and is just as significant. It is not just a curious similarity—the relationship has deep implications in both hyperbolic geometry and the theory of complex numbers. Knowing identities like this helps students solve a variety of problems more efficiently and develop a deeper understanding of the concepts in advanced mathematics.