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Chapter 3: Problem 11

Prove the identity. $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$

Short Answer

Expert verified
The identity is proved by expanding both sides using exponential definitions and showing they are equal.

Step by step solution

01

Recall Definitions

Recall the definitions of hyperbolic sine and cosine. The hyperbolic sine is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \) and the hyperbolic cosine is defined as \( \cosh x = \frac{e^x + e^{-x}}{2} \).
02

Use Definition to Expand Left Side

Start by expanding \( \sinh(x+y) \) using its definition: \( \sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} \).
03

Simplify the Exponential Expressions

Rewrite \( e^{x+y} \) and \( e^{-(x+y)} \) as \( e^x e^y \) and \( e^{-x} e^{-y} \) respectively. This gives \( \sinh(x+y) = \frac{e^x e^y - e^{-x} e^{-y}}{2} \).
04

Substitute Definitions on Right Side

Now expand \( \sinh x \cosh y + \cosh x \sinh y \) using definitions: \( \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} + \frac{e^x + e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2} \). This results in two terms to expand separately.
05

Expand Each Term

Expand each product: \( \frac{1}{4} [(e^x e^y + e^x e^{-y} - e^{-x} e^y - e^{-x} e^{-y}) + (e^x e^y - e^x e^{-y} + e^{-x} e^y - e^{-x} e^{-y})] \).
06

Simplify Product Combination

Combine and simplify the terms: \( \frac{1}{4} [2e^x e^y - 2e^{-x} e^{-y}] \) which simplifies to \( \frac{1}{2} (e^x e^y - e^{-x} e^{-y}) \).
07

Compare Simplified Expressions

Notice that the expanded form from Step 3 matches the form obtained in Step 6. Therefore, \( \sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Sine
The hyperbolic sine function, known as \( \sinh x \), might sound complex and mathematical, but let's think of it as a unique curve that grows and spreads softly like a hyperbola. It's formulated through exponential functions:
  • The definition is \( \sinh x = \frac{e^x - e^{-x}}{2} \).
  • This essentially means you subtract the inverse exponential function from the exponential function and divide by 2.
Hyperbolic sine is analogous to traditional sine in trigonometry, but with distinct characteristics. While trigonometric sine oscillates between -1 and 1, hyperbolic sine grows without bounds. It's particularly useful because it appears in various areas of math, from differential equations to complex analysis. To visualize it, imagine a line stretching upwards and downwards from the origin, increasingly far apart as they move further from zero.
Hyperbolic Cosine
The hyperbolic cosine function, \( \cosh x \), is another fascinating sibling in the hyperbolic family. It has its own unique properties and formula:
  • Defined by \( \cosh x = \frac{e^x + e^{-x}}{2} \).
  • This is simply the average of the exponential function and its inverse.
Unlike its oscillating trigonometric cousin, the hyperbolic cosine never dips below 1, forming a smooth, U-shaped curve. This function also grows exponentially, which is handy for describing many natural phenomena, like the shape of a hanging cable (catenary). Whether it's used in physics or engineering, the hyperbolic cosine is our go-to function for describing symmetrical shapes.
Exponential Functions
Exponential functions are pivotal in understanding hyperbolic functions. Simply put, an exponential function is expressed as \( e^x \), where \( e \) is a constant approximately equal to 2.718.
  • These functions are known for their rapid growth.
  • Anything multiplied by an exponential function can change drastically over a short interval.
They form the backbone of many natural laws and financial models, illustrating growth or decay.
In hyperbolic functions, exponential terms like \( e^x \) and \( e^{-x} \) provide the building blocks, encoding the characteristic properties of \( \sinh x \) and \( \cosh x \) inside their definitions.
By grasping exponential functions, you unlock a key component to understanding how hyperbolic sine and cosine work harmoniously, serving as useful tools across various scientific fields.

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Most popular questions from this chapter

A particle is moving along the curve \(y=\sqrt{x}\) . As the particle passes through the point \((4,2),\) its x-coordinate increases at a rate of 3 \(\mathrm{cm} / \mathrm{s}\) . How fast is the distance from the particle to the origin changing at this instant?

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve \(y=f(x)\) that satisfies the differential equation $$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\rho g}{\mathrm{T}} \sqrt{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}}$$ where \(\rho\) is the linear density of the cable, \(g\) is the acceleration due to gravity, and T is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function $$y=f(x)=\frac{T}{\rho g} \cosh \left(\frac{\rho g x}{T}\right)$$ is a solution of this differential equation.

Newton's Law of Gravitation says that the magnitude \(F\) of the force exerted by a body of mass \(m\) on a body of mass \(M\) is $$\mathrm{F}=\frac{\mathrm{GmM}}{\mathrm{r}^{2}}$$ where \(G\) is the gravitational constant and \(r\) is the distance between the bodies. (a) Find dF/dr and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 \(\mathrm{N} / \mathrm{km}\) when \(\mathrm{r}=20,000 \mathrm{km} .\) How fast does this force change when \(\mathrm{r}=10,000 \mathrm{km}\) ?

\(19-22\) Compute \(\Delta y\) and dy for the given values of \(x\) and \(d x=\Delta x\) . Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and \(\Delta y\) . $$ y=2 x-x^{2}, x=2, \quad \Delta x=-0.4 $$

Show that if \(a \neq 0\) and \(b \neq 0,\) then there exist numbers \(\alpha\) and \(\beta\) such that \(a e^{x}+b e^{-x}\) equals either \(\alpha \sinh (x+\beta)\) or \(\alpha \cosh (x+\beta) .\) In other words, almost every function of the form \(f(x)=a e^{x}+b e^{-x}\) is a shifted and stretched hyperbolic sine or cosine function.
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