What are hyperbolic functions like sinh x, cosh x and tanh x, and how are they defined?

In one line

Hyperbolic functions are built from \(e^x\): \(\sinh x=\dfrac{e^x-e^{-x}}2\), \(\cosh x=\dfrac{e^x+e^{-x}}2\), \(\tanh x=\dfrac{\sinh x}{\cosh x}\); they behave like sine and cosine but without the \(i\).

Explained simply

Hang a chain or a power line between two poles. The curve it makes is not a parabola; it is \(y=a\cosh(x/a)\), called a catenary. That is the most famous real-life appearance of hyperbolic functions. The name comes from the fact that the point \((\cosh t,\sinh t)\) moves on the hyperbola \(x^2-y^2=1\), just as \((\cos t,\sin t)\) moves on the circle \(x^2+y^2=1\).

Definitions (for real or complex \(x\)): \(\sinh x=\dfrac{e^x-e^{-x}}2\), \(\cosh x=\dfrac{e^x+e^{-x}}2\), \(\tanh x=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}\), \(\coth x=\dfrac1{\tanh x}\), \(\mathrm{sech}\,x=\dfrac1{\cosh x}\), \(\mathrm{cosech}\,x=\dfrac1{\sinh x}\).

How it works

1. Basic values: \(\sinh0=0\), \(\cosh0=1\), \(\tanh0=0\).
2. Symmetry: \(\cosh(-x)=\cosh x\) (even), \(\sinh(-x)=-\sinh x\) (odd), \(\tanh(-x)=-\tanh x\).
3. Range for real \(x\): \(\cosh x\ge1\), \(\sinh x\) takes all real values, \(-1<\tanh x<1\).
4. Adding the definitions: \(\cosh x+\sinh x=e^x\); subtracting: \(\cosh x-\sinh x=e^{-x}\).
5. Multiplying these two gives the key identity \(\cosh^2x-\sinh^2x=1\).
6. Derivatives: \(\dfrac{d}{dx}\sinh x=\cosh x\), \(\dfrac{d}{dx}\cosh x=\sinh x\) (no minus sign, unlike \(\cos\)).

Key formula

\(\sinh x=\dfrac{e^x-e^{-x}}2,\quad\cosh x=\dfrac{e^x+e^{-x}}2,\quad\tanh x=\dfrac{e^{2x}-1}{e^{2x}+1}\)

\(x\) is a pure number (not an angle in degrees), \(e\approx2.718\).

Quick example

Find \(\cosh(\log2)\) and \(\tanh(\log2)\) (log means natural log). Since \(e^{\log2}=2\) and \(e^{-\log2}=\tfrac12\):

\(\cosh(\log2)=\dfrac{2+\frac12}{2}=\dfrac54\), \(\ \sinh(\log2)=\dfrac{2-\frac12}2=\dfrac34\), \(\ \tanh(\log2)=\dfrac{3/4}{5/4}=\dfrac35\).

Check: \(\cosh^2-\sinh^2=\dfrac{25}{16}-\dfrac9{16}=1\). Correct.

Common mistakes

• Writing \(\cosh^2x+\sinh^2x=1\) by copying the trig identity; the correct one has a minus sign.
• Thinking \(\sinh x\) is "\(\sin\) of \(hx\)"; it is a separate function read "shine" or "sinch".
• Putting \(x\) in degrees on a calculator; hyperbolic functions take plain numbers.

How MU asks it

Asked in 3-mark Q.1 parts such as "If \(\tanh x=\dfrac12\), find \(\sinh2x\) and \(\cosh2x\)" or "Prove that \((\cosh x+\sinh x)^n=\cosh nx+\sinh nx\)" (since the bracket is \(e^x\)). Write the exponential definitions first, then simplify. Showing the definitions earns the method marks even if arithmetic slips.