Problem 10 - Essential Calculus

Question

Verify the identity. \(\sinh 2 x=2 \sinh x \cosh x\)

Step-by-Step Solution

Answer

After expressing the hyperbolic functions in terms of exponential functions and simplifying, you find that both sides of the equation are indeed equal, thus verifying the given identity.

Step 1: Express the function sinh 2x in terms of the exponential function

Write sinh 2x in terms of exponentials using the definition \(\sinh x = \frac{e^x - e^{-x}}{2}\). Doing this gives: \(\sinh 2x = \frac{e^{2x} - e^{-2x}}{2}\)

Step 2: Express the function sinh x and cosh x in terms of the exponential function

The definitions of sinh x and cosh x, in terms of exponents, are given by: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\), respectively. Multiply these two expressions to get 2 sinh x cosh x: \(2 \sinh x \cosh x = 2 \left(\frac{e^x - e^{-x}}{2}\right)\left(\frac{e^x + e^{-x}}{2}\right) = \frac{e^{2x} - e^{-2x}}{2}\).

Step 3: Compare the expressions obtained in steps 1 and 2

Upon comparing; \(\sinh 2x = \frac{e^{2x} - e^{-2x}}{2}\) and \(2 \sinh x \cosh x = \frac{e^{2x} - e^{-2x}}{2}\), the identity righteousness is evident. Hence the identity is true and is thus confirmed.

Key Concepts

Exponential Functions

Exponential functions play a key role in defining hyperbolic functions. These functions are characterized by expressions involving powers of the mathematical constant \(e\), approximately equal to 2.71828. In the context of the original exercise, the hyperbolic sine and cosine functions, \(\sinh x\) and \(\cosh x\), are expressed in terms of exponential functions.

• The hyperbolic sine, \(\sinh x\), is defined as \(\frac{e^x - e^{-x}}{2}\).
• The hyperbolic cosine, \(\cosh x\), is defined as \(\frac{e^x + e^{-x}}{2}\).

These definitions enable us to express complex relationships between hyperbolic functions using algebraic manipulation of exponential functions. In the original exercise, the identity \(\sinh 2x = 2 \sinh x \cosh x\) is verified using these exponential definitions.

Mathematical Identities

Mathematical identities are equations that are true for all values of the variables involved. For hyperbolic functions, there exist several important identities, similar to those for trigonometric functions. The identity \(\sinh 2x = 2 \sinh x \cosh x\) is one such identity. It reflects the close relationship between the hyperbolic sine and cosine functions.

These identities are not just mathematical curiosities; they are essential tools in calculus and engineering.

• Identities help simplify expressions and solve complex equations.
• They are used in integration and differentiation of functions.

Verifying an identity, as done in the exercise, involves proving both sides of the equation are equal for all permissible values of \(x\). This can be achieved through substitution, similar to the step-by-step approach used in demonstrating \(\sinh 2x = 2 \sinh x \cosh x\).

Algebraic Manipulations

Algebraic manipulations involve rewriting and rearranging expressions to achieve a desired form. This skill is vital in solving mathematical problems and proving identities. In the exercise, we used algebraic manipulations to show that \(\sinh 2x\) and \(2 \sinh x \cosh x\) are equivalent.

• First, we expanded \(\sinh 2x\) using its exponential definition, which was \(\frac{e^{2x} - e^{-2x}}{2}\).
• Next, we expanded \(2 \sinh x \cosh x\) by multiplying their exponential expressions which simplified to the same form.

By equating these forms, we demonstrated that the identity holds true. These steps highlight how understanding the underlying relationships and performing precise algebraic operations can lead to solving complex problems.