Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Verify that the given equations are identities. \(\sinh 2 x=2 \sinh x \cosh x\)

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Recall Definitions

The hyperbolic sine function is defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\), and the hyperbolic cosine function is defined as \(\cosh x = \frac{e^x + e^{-x}}{2}\). These definitions will be used to expand both sides of the equation \(\sinh 2x \text{ and } 2\sinh x\cosh x\).

Step 2: Expand Left Side

Use the identity \(\sinh(2x) = 2\sinh x\cosh x\) directly, or for a full derivation, apply the hyperbolic angle addition formulas. \(\sinh(2x) = \frac{e^{2x} - e^{-2x}}{2}\).

Step 3: Expand Right Side

Calculate \(2 \sinh x \cosh x = 2 \cdot \frac{e^x - e^{-x}}{2} \cdot \frac{e^x + e^{-x}}{2} = \frac{1}{2} (e^{2x} + e^{-2x} - e^x \cdot e^{-x} + e^{-x} \cdot e^x) = \frac{1}{2} (e^{2x} - e^{-2x})\).

Step 4: Compare Expressions

The expression \(\frac{1}{2} (e^{2x} - e^{-2x})\) derived in Step 3 is equivalent to the expansion of \(\sinh(2x) = \frac{e^{2x} - e^{-2x}}{2}\) derived in Step 2. Since both sides are equal, the given equation \(\sinh 2x = 2 \sinh x \cosh x\) is verified as an identity.

Key Concepts

sinh function

The hyperbolic sine function, commonly notated as \(\sinh x\), is one of the fundamental hyperbolic functions. It shares many analogies with the sine function encountered in circular trigonometry but applies to hyperbolic geometry. Defined as:

• \(\sinh x = \frac{e^x - e^{-x}}{2}\)

This definition uses the natural exponential function to create a relationship that is applicable in various mathematical and physical contexts. Hyperbolic sine is particularly useful in calculations involving hyperbolic trajectories and areas.'sinh x' increases exponentially with positive \(x\) and decreases exponentially with negative \(x\).

To visualize \(\sinh x\), imagine tracing a curve that balances between the exponential growth and decay, reflecting its symmetrical nature about the origin.

cosh function

The hyperbolic cosine function, symbolized as \(\cosh x\), pairs with the \(\sinh x\) function. Just as with the \(\sinh x\), \(\cosh x\) is intimately tied to nature's exponential functions. It is defined as:

• \(\cosh x = \frac{e^x + e^{-x}}{2}\)

Despite working similarly to the cosine in trigonometry, \(\cosh x\) distinguishes itself with its exponential rise, maintaining positivity for all \(x\).

The graph of \(\cosh x\) takes on the form of a softly rising curve with a minimum at \(x=0\). This characteristic is due to the addition of the exponential terms. Because of this, \(\cosh x\) remains greater than or equal to one, providing stability in hyperbolic equations.

mathematical identities

Mathematical identities are foundational truths in mathematics. They state that two formulations are equivalent for all permissible values of the variables involved. In studying hyperbolic functions, certain identities can simplify complex expressions. For example, the identity we've examined states:

• \(\sinh 2x = 2 \sinh x \cosh x\)

Identities like this are crucial as they allow mathematicians to equate seemingly different expressions, revealing underlying symmetries and connections. Verifying identities often involves replacing functions with their exponential definitions and simplifying to show equality. By proving these relationships, one gains insight into the interconnected nature of mathematical concepts.

Mastering these identities can tremendously aid in solving equations and calculus problems involving hyperbolic functions.

hyperbolic angle addition

The hyperbolic angle addition formulas are essential tools in verifying and manipulating expressions involving hyperbolic functions. Much like their trigonometric counterparts, these formulas allow for the simplification of the sums and differences of hyperbolic angles:

• \(\sinh(a + b) = \sinh a \cosh b + \cosh a \sinh b\)
• \(\cosh(a + b) = \cosh a \cosh b + \sinh a \sinh b\)

These formulas enable the conversion of complex arguments into combinations of simpler terms, opening pathways to verification and solution of various identities and equations. They are especially useful in handling problems involving hyperbolic functions of multiple arguments, as demonstrated in verifying identities like \(\sinh 2x = 2 \sinh x \cosh x\).

Understanding and applying hyperbolic angle addition can transform complicated hyperbolic function evaluations into straightforward calculations.