The hyperbolic sine and hyperbolic cosine functions from ℝ to ℝ are defined by sinhx:=(e^x-e^-x)

The hyperbolic sine and hyperbolic cosine functions from ℝ...

The hyperbolic sine and hyperbolic cosine functions from $\mathbb{R}$ to $\mathbb{R}$ are defined by $$ \sinh x:=\frac{e^{x}-e^{-x}}{2} \quad \text { and } \quad \cosh x:=\frac{e^{x}+e^{-x}}{2} \quad \text { for } x \in \mathbb{R} . $$ Show that for any $t \in \mathbb{R}$, the point $(\cosh t, \sinh t)$ is on the hyperbola $x^{2}-y^{2}=1$. Also, show that (i) $\sinh 0=0, \cosh 0=1$ and $\cosh ^{2}-\sinh ^{2}=1$ for all $x \in \mathbb{R}$. (ii) $(\sinh )^{\prime} x=\cosh x$ and $(\cosh )^{\prime} x=\sinh x$ for all $x \in \mathbb{R}$. (iii) $\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$ and $\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y$ for all $x, y \in \mathbb{R}$ Sketch the graphs of the functions sinh and cosh.

The hyperbolic sine and hyperbolic cosine functions from $\mathbb{R}$ to $\mathbb{R}$ are defined by
$$
\sinh x:=\frac{e^{x}-e^{-x}}{2} \quad \text { and } \quad \cosh x:=\frac{e^{x}+e^{-x}}{2} \quad \text { for } x \in \mathbb{R} .
$$
Show that for any $t \in \mathbb{R}$, the point $(\cosh t, \sinh t)$ is on the hyperbola $x^{2}-y^{2}=1$. Also, show that
(i) $\sinh 0=0, \cosh 0=1$ and $\cosh ^{2}-\sinh ^{2}=1$ for all $x \in \mathbb{R}$.
(ii) $(\sinh )^{\prime} x=\cosh x$ and $(\cosh )^{\prime} x=\sinh x$ for all $x \in \mathbb{R}$.
(iii) $\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$ and
$\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y$ for all $x, y \in \mathbb{R}$
Sketch the graphs of the functions sinh and cosh.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh and cosh, are analogues of the classical trigonometric functions but are defined using exponential functions. They play a central role in various areas of mathematics, including calculus, differential equations, and hyperbolic geometry, often modeling unstretched hyperbolic symmetry rather than circular symmetry.

Exponential Function Definitions

The definitions of sinh and cosh employ the exponential function, with sinh x defined as (e^x - e^(-x))/2 and cosh x defined as (e^x + e^(-x))/2. These definitions connect hyperbolic functions to growth and decay processes, and provide a pathway to derive their many properties, such as differentiability and the hyperbola relation.

Hyperbola

The hyperbola x^2 - y^2 = 1 represents a conic section, akin to how the circle x^2 + y^2 = 1 relates to trigonometric functions. When the point (cosh t, sinh t) is considered, it naturally lies on this hyperbola, establishing an important geometric interpretation of hyperbolic functions and linking them to hyperbolic geometry.

Derivatives of Hyperbolic Functions

A key concept in calculus is the differentiation of hyperbolic functions. The derivative of sinh x is cosh x and vice versa for cosh x (with an appropriate sign consideration in some contexts), which is analogous to the differentiation properties of sine and cosine. This property is fundamental in solving differential equations where hyperbolic functions appear as solutions.

Hyperbolic Identities and Addition Formulas

Hyperbolic functions satisfy various identities that mirror those for trigonometric functions. Notably, cosh^2 x - sinh^2 x = 1 is analogous to the Pythagorean identity for sine and cosine. Additionally, the addition formulas, such as sinh(x+y) = sinh x cosh y + cosh x sinh y and cosh(x+y) = cosh x cosh y + sinh x sinh y, are instrumental in simplifying expressions and solving equations involving hyperbolic functions.

Graphical Representation of Hyperbolic Functions

The graphs of hyperbolic sine and cosine functions exhibit distinctive shapes; cosh x forms a U-shaped curve centered at the y-axis with a minimum at x = 0, while sinh x is an odd function that increases steadily, reflecting its symmetry about the origin. Graphical interpretations help visualize the exponential growth behavior and the non-periodic nature of these functions, contrasting them with their trigonometric counterparts.

Transcript

00:01 This question introduces a couple of new functions and asks you to use exponents to prove an equation.

00:08 The functions that it introduces are hyperbolic cosine, which you write cosh, which is equal to e to the x plus e to the negative x over 2.

00:23 The other one is hyperbolic sine, which you write sine h of x, which is equal to e to the x minus e to the negative x.

00:34 X over 2.

00:37 Let me fix this up.

00:41 So this question asks you to prove that kos h of x squared minus sine h of x squared equals 1.

00:59 Well let me start by figuring out each bit individually.

01:07 Kos h of x squared is going to look like this.

01:12 E to the x plus e to the negative x over 2 squared.

01:19 We know when we square fractions, we square the numerator and the denominator.

01:24 So what we'll end up with is e to the x plus e to the negative x squared over two squared, which is four.

01:33 And when we use foil to square this binomial, we'll be left with e to the x squared plus two, e to the x times e to the minus x plus e to the minus x squared.

01:56 All of that is over four.

01:59 Now here's something interesting.

02:01 We have e to the x times e to the minus x.

02:04 What is that? well, we know that e to the minus x is the same thing as 1 over e to the x.

02:15 Remember when we have a negative exponent, it moves it to the denominator...

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