Summary
- sinh x, cosh x, tanh x are defined directly from ex and e−x — no circles involved.
- The three reciprocal functions cosech x, sech x, coth x behave like cosec, sec, cot.
- Hyperbolic functions obey addition/double-angle formulae that look almost identical to trig ones.
- Osborne's Rule is the shortcut: convert any trig identity to a hyperbolic one, flipping the sign in front of any term that is (or hides) a product of two sines.
- The inverse functions arsinh, arcosh, artanh undo sinh, cosh, tanh, and each has a closed logarithmic form.
- Equations mixing sinh/cosh can be solved either by converting to exponentials, or by using hyperbolic identities + the log form of the inverse.
1. Definitions and Graphs
Forget circles for a second. Hyperbolic functions are just combinations of ex and
e−x, built to mimic the shape and algebra of sin, cos and tan without actually
being trigonometric at all. That's why they're called "hyperbolic" — they relate to the hyperbola
x² − y² = 1 the way sin/cos relate to the circle x² + y² = 1.
Because they're built from exponentials, you can read off their behaviour instantly:
- sinh x is an odd function (sinh(−x) = −sinh x), passes through the origin, and grows like ½eˣ for large x.
- cosh x is an even function (cosh(−x) = cosh x), has a minimum of 1 at x = 0 — it never dips below 1 — and looks like a smile (this is the actual shape of a hanging chain, a "catenary").
- tanh x squeezes everything between −1 and 1: it flattens out to horizontal asymptotes y = 1 and y = −1 as x → ±∞.
y = sinh x → odd, S-shaped, passes through (0,0), no asymptotes
y = cosh x → even, U-shaped, minimum point (0,1), never negative
y = tanh x → odd, S-shaped, bounded between y = −1 and y = 1
The reciprocal functions
Exactly as sec = 1/cos, cosec = 1/sin, cot = 1/tan, we define:
Their graphs follow directly from "flipping" sinh, cosh, tanh:
- cosech x has a vertical asymptote at x = 0 (since sinh 0 = 0) and two separate branches, one above and one below the x-axis.
- sech x has a maximum value of 1 at x = 0 (since cosh x ≥ 1 always) and tends to 0 as x → ±∞.
- coth x has a vertical asymptote at x = 0, and horizontal asymptotes at y = 1 and y = −1.
Quick sanity check
You should be able to sketch all six graphs from memory in under a minute — examiners love asking
for a quick sketch as a "free" first mark. The shape of cosech/sech/coth always follows from the
shape of sinh/cosh/tanh: asymptotes appear wherever the original function was zero, and vice versa.
Practice: Show that cosh²x − sinh²x = 1 directly from the exponential definitions.
2. Addition Formulae, Double Angle Formulae, and Osborne's Rule
3. Inverse Hyperbolic Functions
Graphs
4. Equations Involving Hyperbolic Functions
There are two main strategies, and picking the right one saves a lot of time:
Worked example — Strategy A