The hyperbolic functions defined from exponentials, their graphs and properties, the key identities, and the logarithmic forms of the inverse hyperbolic functions.

What this dot point is asking

OCR wants you to define the hyperbolic functions $\sinh$, $\cosh$ and $\tanh$ in terms of exponentials, sketch and describe their graphs (including odd or even symmetry and asymptotes), use the key identities (above all $\cosh^2 x - \sinh^2 x = 1$), and derive and use the logarithmic forms of the inverse hyperbolic functions.

Definitions from exponentials

The hyperbolic functions are combinations of $e^x$ and $e^{-x}$, named by analogy with the circular (trigonometric) functions. They appear naturally in calculus, in the catenary (a hanging chain) and in solutions of differential equations.

Graphs and properties

The graphs follow from the definitions. $\cosh x$ is even (symmetric about the $y$-axis), always $\ge 1$, with a minimum at $(0, 1)$ and both arms rising like $\tfrac{1}{2}e^{|x|}$. $\sinh x$ is odd, passing through the origin and increasing throughout. $\tanh x$ is odd, increasing, sandwiched between the horizontal asymptotes $y = 1$ and $y = -1$.

The key identities

The hyperbolic identities mirror the trigonometric ones, with sign changes captured by Osborn's rule. The central one follows directly from the definitions.

The inverse hyperbolic functions

Because the hyperbolic functions are built from exponentials, their inverses are logarithms. You derive each by setting $y$ equal to the inverse, writing the defining exponential equation, and solving the resulting quadratic in $e^y$. The same three-step pattern works every time: write $x = \sinh y$ (or $\cosh y$, or $\tanh y$) in exponential form, multiply through by a power of $e^y$ to obtain a quadratic in $e^y$, then solve and take logarithms, rejecting any root that would make $e^y$ negative. Memorising the final logarithmic forms is useful for speed, but you must be able to derive them on demand, because OCR sometimes asks for the derivation as a "show that" question worth several marks.

Hyperbolic functions feed directly into the calculus topic, where their derivatives and integrals (and the inverse forms as standard integrals) are examined.

Try this

Q1. State the value of $\sinh 0$ and $\cosh 0$. [2 marks]

• Cue. $\sinh 0 = \dfrac{1 - 1}{2} = 0$; $\cosh 0 = \dfrac{1 + 1}{2} = 1$.

Q2. Use the identity to find $\cosh^2 x$ if $\sinh x = \tfrac{3}{4}$. [2 marks]

• Cue. $\cosh^2 x = 1 + \sinh^2 x = 1 + \tfrac{9}{16} = \tfrac{25}{16}$.