Problem 13 - Calculus of a Single Variable

Question

In Exercises 7–14, verify the identity. $$ \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y $$

Step-by-Step Solution

Answer

The given identity \(\sinh (x+y) = \sinh x \cdot \cosh y + \cosh x \cdot \sinh y\) holds true.

Step 1: Rewrite \(\sinh (x+y)\) in terms of exponentials

Using the definition of hyperbolic sine function, \(\sinh (x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2}\). So, the left side of the identity becomes \( \frac{e^{x+y} - e^{-(x+y)}}{2}\).

Step 2: Rewrite \(\sinh x \cdot \cosh y + \cosh x \cdot \sinh y\) in terms of exponentials

Using the definition of hyperbolic sine and hyperbolic cosine functions, \(\sinh x \cdot \cosh y + \cosh x \cdot \sinh y = \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} + \frac{e^x + e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2}\). This simplifies to \(\frac{e^{x+y} - e^{-(x+y)}}{2}\).

Step 3: Compare the results of Step 1 and Step 2

The result from Step 1 and Step 2 are identical: \( \frac{e^{x+y} - e^{-(x+y)}}{2} = \frac{e^{x+y} - e^{-(x+y)}}{2}\). Therefore, the given identity \(\sinh (x+y) = \sinh x \cdot \cosh y + \cosh x \cdot \sinh y\) holds true.

Key Concepts

sinh function

The hyperbolic sine function, represented as \( \sinh x \), is an important function in mathematics, especially within calculus and complex analysis. Unlike the more commonly known sine function, the hyperbolic sine does not oscillate. It is, in fact, related to the exponential function.
The formula for the hyperbolic sine function is given by:

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

The expression combines two exponentials: \( e^x \) and \( e^{-x} \). From this, you can see that \( \sinh x \) is always increasing and grows exponentially.

Hyperbolic functions, including \( \sinh x \), are often encountered in the solution of linear differential equations and in describing the shapes of cables or arcs, among other applications. Knowing the exponential identity helps in verifying complex equations and transformations in hyperbolic functions.

cosh function

Hyperbolic cosine, denoted as \( \cosh x \), is another fundamental function in the realm of hyperbolic functions. It also connects deeply with the exponential function and is used to model real-world phenomena like the shape of a catenary.
The formula for \( \cosh x \) is:

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

Notice how similar it looks to \( \sinh x \), but it uses a plus sign instead of a minus. Like the hyperbolic sine, \( \cosh x \) does not oscillate and rises from 1. The function is always greater than or equal to one and increases exponentially.

This hyperbolic function is even, meaning \( \cosh(-x) = \cosh x \), which differentiates it from \( \sinh x \). Understanding \( \cosh x \) and its properties is essential as it often appears in equations involving the sinh function, adding to the versatility of solving complex mathematical problems.

exponential identities

Exponential identities form the core relationships that underpin hyperbolic functions like \( \sinh x \) and \( \cosh x \). These identities make it possible to express these functions purely in terms of exponentials, which in turn simplifies many mathematical problems.
Let's recap the exponential expressions for \( \sinh x \) and \( \cosh x \):

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

These representations are powerful because they allow algebraic manipulation using exponential rules, like the exponent addition property. When you see an identity like \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \), you can rewrite each component with exponentials to verify the identity land manipulate equations effectively.

Exponential identities are crucial in solving problems involving hyperbolic functions, allowing transformations and simplifications that align with broader mathematical expressions and models. They establish a bridge between polynomial and exponential behavior, aiding in the understanding of more complex relationships in mathematics.