4.07a–4.07c - Hyperbolic definitions, graphs and identities | OCR A-Level Further Mathematics A

Build three new functions from exponentials: $\sinh x$ , $\cosh x$ and $\tanh x$ . You will use their definitions and domains and ranges in 4.07a, sketch their graphs in 4.07b, and prove and use identities in 4.07c, including solving equations exactly.

Building the functions from exponentials

Can two ordinary exponential curves combine to trace a hyperbola? Start with $x = \ln 2$ : then $e^x = 2$ and $e^{-x} = \frac12$ . Half their sum is $\frac54$ , and half their difference is $\frac34$ . These two numbers satisfy

\left(\frac54\right)^2 - \left(\frac34\right)^2 = 1

By the end of the lesson you will see why this happens for every real $x$ , not just this convenient choice.

You already know

For real $x$ , $e^x > 0$ , $e^{-x} = 1/e^x$ , and $e^{a+b} = e^a e^b$ . Also, $e^{\ln a} = a$ when $a > 0$ .

Adding and subtracting the same two exponentials creates the hyperbolic cosine and hyperbolic sine. Their names are pronounced “cosh” and “shine”, respectively; the letter $h$ stays in the notation throughout your working.

Hyperbolic functions

For real $x$ ,

\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}

For example, $\cosh 0 = 1$ and $\sinh 0 = \tanh 0 = 0$ . These are different functions from $\cos x$ , $\sin x$ and $\tan x$ .

The denominator of $\tanh x$ never vanishes, because $e^x + e^{-x} > 0$ . Multiplying its numerator and denominator by $e^x$ gives a useful second form.

Exponential definitions

For every real $x$ ,

\sinh x = \frac{e^x-e^{-x}}{2}, \qquad \cosh x = \frac{e^x+e^{-x}}{2}, \qquad \tanh x = \frac{e^{2x}-1}{e^{2x}+1}

Not in the formula booklet: learn it.

When $x$ is a logarithm, evaluate the two exponentials first. The resulting fractions are often much simpler than a decimal approximation suggests.

Worked example 1: exact values at a logarithm

Question. Find the exact values of $\sinh(\ln 3)$ , $\cosh(\ln 3)$ and $\tanh(\ln 3)$ .

Plan. Substitute $e^{\ln 3} = 3$ and $e^{-\ln 3} = \frac13$ into the definitions, then divide.

\begin{aligned} \sinh(\ln 3) &= \frac{3-\frac13}{2} = \frac43 && \text{half the difference} \\ \cosh(\ln 3) &= \frac{3+\frac13}{2} = \frac53 && \text{half the sum} \\ \tanh(\ln 3) &= \frac{4/3}{5/3} = \frac45 && \sinh x/\cosh x \end{aligned}

Why this step? A negative exponent means a reciprocal; it does not put a minus sign in front of $3$ .

Answer. $\frac43$ , $\frac53$ and $\frac45$ , respectively.

Check. The alternative form gives $\tanh(\ln 3) = (9-1)/(9+1) = \frac45$ .

The next example changes just the sign of the input. Watch which output changes sign: this predicts the symmetries of the graphs.

Worked example 2: a negative input

Question. Find the exact values of $\sinh(-\ln 3)$ , $\cosh(-\ln 3)$ and $\tanh(-\ln 3)$ .

Plan. Interchange the exponential values: now $e^x = \frac13$ and $e^{-x} = 3$ .

\begin{aligned} \sinh(-\ln 3) &= \frac{\frac13-3}{2} = -\frac43 && \text{difference reverses} \\ \cosh(-\ln 3) &= \frac{\frac13+3}{2} = \frac53 && \text{sum unchanged} \\ \tanh(-\ln 3) &= \frac{-4/3}{5/3} = -\frac45 && \text{divide} \end{aligned}

Why this step? Replacing $x$ by $-x$ swaps $e^x$ and $e^{-x}$ ; a sum survives the swap, but a difference changes sign.

Answer. $-\frac43$ , $\frac53$ and $-\frac45$ , respectively.

Check. $\cosh(-\ln 3)$ is still positive, as a sum of positive exponentials must be.

The sign change is predictable, so a negative hyperbolic cosine is a useful warning that a calculation has gone wrong.

Watch out

Wrong: $\cosh(-x) = -\cosh x$ . The exponential terms swap places in a sum, so $\cosh(-x) = \cosh x$ . It is $\sinh(-x) = -\sinh x$ and $\tanh(-x) = -\tanh x$ that change sign.

Try the same calculation with a different logarithm; retain exact fractions.

Graphs, domains and ranges

The graphs explain which equations have real solutions before you start any algebra.

[DIAGRAM: asset_name: The three hyperbolic graphs; asset_slug: h245_hyperbolic_graphs; caption: Hyperbolic sine is odd and unbounded, hyperbolic cosine is even with minimum 1, and hyperbolic tangent lies strictly between -1 and 1; description: Three coordinate panels show y = sinh x through (0, 0), y = cosh x with minimum (0, 1), and y = tanh x through (0, 0). The first two panels include pale dashed exponential component curves e to x divided by 2 and plus or minus e to minus x divided by 2. The tanh panel has dashed horizontal asymptotes y = 1 and y = -1. All three panels are labelled with their domains and ranges]
Hyperbolic sine is odd and unbounded, hyperbolic cosine is even with minimum 1, and hyperbolic tangent lies strictly between -1 and 1

For $\sinh x$ , increasing $x$ makes $e^x$ larger and $e^{-x}$ smaller, so their difference increases strictly. It tends without bound in both directions. Thus each horizontal line crosses the sine graph exactly once.

For $\cosh x$ , set $t = e^x > 0$ . Then

\cosh x - 1 = \frac{t+t^{-1}-2}{2} = \frac{(t-1)^2}{2t} \ge 0

Equality holds only when $t = 1$ , so only when $x = 0$ . The graph decreases to $(0,1)$ , then increases symmetrically. A line $y = k$ therefore crosses it twice for $k > 1$ , once for $k = 1$ , and never for $k < 1$ .

For $\tanh x$ , put $u = e^{2x} > 0$ . The expression $(u-1)/(u+1) = 1-2/(u+1)$ increases as $u$ increases. It approaches $1$ as $x \to \infty$ and $-1$ as $x \to -\infty$ , but reaches neither value.

Domain and range

The domain is the set of allowed inputs; the range is the set of outputs actually attained. For the real hyperbolic functions:

Function	Domain	Range	Symmetry
$\sinh x$	$\mathbb R$	$\mathbb R$	odd
$\cosh x$	$\mathbb R$	$[1,\infty)$	even
$\tanh x$	$\mathbb R$	$(-1,1)$	odd

For example, $1$ belongs to the range of $\cosh$ , but not to the range of $\tanh$ .

For a sketch, label the intercept, the symmetry and any asymptotes, then draw the correct end behaviour. Hyperbolic functions have no repeating periods: these curves do not oscillate like their trigonometric namesakes. The domain/range facts and sketch features are not in the formula booklet; learn them.

Key idea

$\cosh x \ge 1$ for every real $x$ ; $-1 < \tanh x < 1$ . Endpoints belong to the cosine range but not to the tangent range.

Use those precise inequalities, rather than a rough picture, to reject impossible values.

Proving and using identities

The functions satisfy identities because their exponential definitions fit together, rather than because you can copy a trigonometric identity and change its name.

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4.07a–4.07c - Hyperbolic definitions, graphs and identities

Building the functions from exponentials

Graphs, domains and ranges

Proving and using identities

Continue this lesson.

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