The hyperbolic functions

Summary

This lesson studies the hyperbolic cosine, sine and tangent, three functions obtained by combining the natural exponential $e^x$ with its reciprocal $e^{-x}$ . It determines their parity, their sign, their monotonicity and their range, and proves the identity $\cosh^2 x - \sinh^2 x = 1$ , according to which the point $\left( \cosh t, \sinh t \right)$ traverses the right branch of the hyperbola $x^2 - y^2 = 1$ and passes exactly once through each of its points, which is where their name comes from. It then derives the addition and double-argument formulas and obtains the inverse functions, which are expressed by means of logarithms. Finally, it describes the catenary, which is the shape of a hanging chain, and the area of the hyperbolic sector, which announces the analogy with the circular functions.

Learning objectives

Define the hyperbolic cosine, sine and tangent from the natural exponential, and prove their parity, their sign, their monotonicity and their bounds.
Prove the fundamental identity and use it to parametrize the branches of hyperbolas, recognizing that each point of a branch corresponds to a unique parameter.
Derive the addition and double-argument formulas, and apply them to proving identities and inequalities.
Obtain the inverse hyperbolic functions with their domains, by solving quadratic equations in $e^y$ , and use them to solve hyperbolic equations.

Definition and first properties

In the lesson on the number e and the natural exponential, the function $e^x$ was defined, positive and strictly increasing on the whole real line. Its reciprocal $e^{-x} = \frac{1}{e^x}$ is also positive, but strictly decreasing, and its graph is the reflection of that of $e^x$ in the $y$ -axis, according to the lesson on function operations. Half the sum and half the difference of these two functions, and their quotient, appear so often in geometry, in physics and in calculus that they receive names of their own.

For every real number $x$ , the hyperbolic cosine, the hyperbolic sine and the hyperbolic tangent of $x$ are defined as half the sum of $e^x$ and $e^{-x}$ , half their difference, and the quotient of the second by the first, respectively.

\cosh x = \frac{e^x + e^{-x}}{2}, \qquad \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}

The three functions are defined on all of $\mathbb{R}$ : the denominator of the tangent is a sum of two positive numbers and never vanishes. At $x = 0$ we have $e^0 = 1$ , so that $\cosh 0 = 1$ , $\sinh 0 = 0$ and $\tanh 0 = 0$ . The adjective “hyperbolic” will be justified in the third section, where it will be seen that these functions describe a hyperbola.

LemmaParity and decomposition of the exponential

The hyperbolic cosine is an even function, and the hyperbolic sine and tangent are odd functions. Moreover, for every real $x$ we have $e^x = \cosh x + \sinh x$ and $e^{-x} = \cosh x - \sinh x$ .

When $x$ is replaced by $-x$ , the functions $e^x$ and $e^{-x}$ exchange their roles. The figure shows the graphs of the hyperbolic cosine and sine with two points of opposite abscissas on each.

Proof

$\cosh(-x) = \frac{\resaltar{e^{-x} + e^{x}}}{2} = \cosh x$
When $x$ is replaced by $-x$ , half the sum does not change, because its two terms are swapped: the hyperbolic cosine is even, and its graph is symmetric about the $y$ -axis.
$\sinh(-x) = \frac{\resaltar{e^{-x} - e^{x}}}{2} = -\sinh x$
Half the difference, on the other hand, changes sign: the hyperbolic sine is odd, and its graph is symmetric about the origin.
$\tanh(-x) = \frac{\resaltar{-\sinh x}}{\resaltar{\cosh x}} = -\tanh x$
The tangent, the quotient of an odd function by an even one, is odd.
$\cosh x + \sinh x = \resaltar{e^x}, \qquad \cosh x - \sinh x = \resaltar{e^{-x}}$
Adding and subtracting the definitions of the hyperbolic cosine and sine recovers the two exponentials: $\cosh x$ and $\sinh x$ are, respectively, the even part and the odd part of the exponential.

ExampleThe graphs of the hyperbolic cosine and sine

The graphs of $\cosh$ and of $\sinh$ are built by adding and subtracting, abscissa by abscissa, the ordinates of $\frac{e^x}{2}$ and of $\frac{e^{-x}}{2}$ .

Proof

$y = \frac{e^x}{2}, \quad \frac{e^0}{2} = \frac{1}{2}$
The graph of $\frac{e^x}{2}$ is obtained from that of $e^x$ by compressing it vertically by half: it crosses the $y$ -axis at $\frac{1}{2}$ , grows without bound to the right and approaches the $x$ -axis to the left.
$y = \frac{e^{-x}}{2}$
The graph of $\frac{e^{-x}}{2}$ is the reflection of the first one in the $y$ -axis.
$\cosh 0 = \resaltar{\frac{1}{2} + \frac{1}{2}} = \resaltar{1}$
At each abscissa, $\cosh x$ is the sum of the two ordinates. At $0$ both equal $\frac{1}{2}$ : the graph passes through $(0, 1)$ , and it is symmetric about the $y$ -axis, because the hyperbolic cosine is even.
$\cosh x - \frac{e^x}{2} = \frac{e^{-x}}{2} > 0$
The hyperbolic cosine exceeds $\frac{e^x}{2}$ by exactly $\frac{e^{-x}}{2}$ , a positive quantity that becomes as small as desired as $x$ grows: to the right, the graph of $\cosh$ approaches that of $\frac{e^x}{2}$ from above, and to the left, by symmetry, that of $\frac{e^{-x}}{2}$ .
$\sinh 0 = \resaltar{\frac{1}{2} - \frac{1}{2}} = \resaltar{0}$
The hyperbolic sine is the difference of the ordinates. It passes through the origin and its graph is symmetric about the origin, because the hyperbolic sine is odd.
$\cosh x - \sinh x = e^{-x} > 0$
To the right, the graph of $\sinh$ approaches that of $\frac{e^x}{2}$ from below, while that of $\cosh$ does so from above: the two get closer to each other without ever touching, because their difference is positive.

The sign of the hyperbolic sine can be read off its definition: $\sinh x > 0$ is equivalent to $e^x > e^{-x}$ , and since the exponential is strictly increasing, this happens exactly when $x > -x$ , that is, when $x > 0$ . Therefore, the hyperbolic sine is negative if $x < 0$ , zero at $0$ and positive if $x > 0$ ; the hyperbolic tangent, the quotient of the sine by a positive number, has the same sign. The hyperbolic cosine, by contrast, is always positive, and something more is true.

LemmaLower bound of the hyperbolic cosine

For every real $x$ we have $\cosh x - 1 = 2 \sinh^2 \frac{x}{2}$ . Consequently, $\cosh x \geq 1$ , and equality holds only when $x = 0$ .

The idea is to recognize in the numerator of $\cosh x - 1$ the square of a difference. The figure shows the graph of the hyperbolic cosine above the line $y = 1$ , with the difference between the two as a vertical line segment.

Proof

$e^x = \left( e^{\frac{x}{2}} \right)^2, \quad e^{-x} = \left( e^{-\frac{x}{2}} \right)^2, \quad \dato{p}{e^{\frac{x}{2}} e^{-\frac{x}{2}} = 1}$
By the laws of exponents, $e^x$ is the square of $e^{\frac{x}{2}}$ , $e^{-x}$ is that of $e^{-\frac{x}{2}}$ , and the product of these two powers is $e^0 = 1$ .
$\cosh x - 1 = \frac{\resaltar{e^x - 2 + e^{-x}}}{2} = \frac{\resaltar{\left( e^{\frac{x}{2}} - e^{-\frac{x}{2}} \right)^2}}{2} = \dato{c}{2 \sinh^2 \frac{x}{2}}$
Therefore, the numerator of $\cosh x - 1$ is the square of a difference, and that difference is twice $\sinh \frac{x}{2}$ .
$\cosh x - 1 = \resaltar{2 \sinh^2 \frac{x}{2}} \geq 0$
A square is not negative: the hyperbolic cosine is at least $1$ .
$\cosh x = 1 \leftrightarrow e^{\frac{x}{2}} = e^{-\frac{x}{2}} \leftrightarrow \resaltar{x = 0}$
Equality requires the base of the square to be zero, that is, $e^{\frac{x}{2}} = e^{-\frac{x}{2}}$ ; since the exponential is injective, this requires $\frac{x}{2} = -\frac{x}{2}$ , that is, $x = 0$ .

The inequality $\cosh x \geq 1$ is the inequality between the arithmetic mean and the geometric mean of the positive numbers $e^x$ and $e^{-x}$ , whose geometric mean is $1$ .

In the rest of this lesson

Monotonicity, range and graphs
The asymptotes of the hyperbolic tangent
The catenary
The hyperbolic functions and the hyperbola
The hyperbolic sector and the name of these functions
Addition formulas
The inverse functions
Solved problems
An equation with two opposite solutions
The tangent of the double argument
The parametrization of a hyperbola
The powers of the sum of the cosine and the sine
The catenary and its parabola
Proposed problems
An inequality between the sine and the tangent
A point of the hyperbola at distance three from the origin
The addition of hyperbolic tangents
Two inverses that coincide
A telescoping sum of hyperbolic cosines