Hyperbolic Functions: Definitions and Identities

Hyperbolic Functions: Definitions and Identities

Hyperbolic functions are the "hyperbolic analogues" of trigonometric functions. Where trig functions parameterise the unit circle, hyperbolic functions parameterise the unit hyperbola x² − y² = 1. They appear throughout further calculus, differential equations, and physics.

Definitions

The hyperbolic functions are defined in terms of exponentials:

Function	Definition	Domain	Range
sinh x	(eˣ − e⁻ˣ)/2	All reals	All reals
cosh x	(eˣ + e⁻ˣ)/2	All reals	[1, ∞)
tanh x	sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)	All reals	(−1, 1)
sech x	1/cosh x	All reals	(0, 1]
cosech x	1/sinh x	x ≠ 0	ℝ \ {0}
coth x	cosh x / sinh x	x ≠ 0	(−∞,−1) ∪ (1,∞)

Key Properties

• sinh x is an odd function: sinh(−x) = −sinh x
• cosh x is an even function: cosh(−x) = cosh x
• tanh x is an odd function: tanh(−x) = −tanh x
• cosh x ≥ 1 for all x, with minimum at x = 0
• tanh x → ±1 as x → ±∞ (horizontal asymptotes)

Graph Shapes

• sinh x: passes through origin, looks like a steeper version of x³ for small x, grows exponentially
• cosh x: U-shaped with minimum (0, 1) — this is the catenary curve (shape of a hanging chain)
• tanh x: S-shaped, passes through origin, bounded between −1 and 1

Osborn's Rule

To convert a trig identity to its hyperbolic version:

1. Replace every trig function with its hyperbolic counterpart (sin → sinh, cos → cosh, etc.)

2. Wherever a product of two sines appears (explicitly or implicitly), change its sign

This means sin²θ → −sinh²x, but cos²θ → cosh²x (no sign change) and sin θ cos θ → sinh x cosh x (only one sine, no sign change).

Fundamental Identities

Trig identity	Hyperbolic analogue
cos²θ + sin²θ = 1	cosh²x − sinh²x = 1
1 + tan²θ = sec²θ	1 − tanh²x = sech²x
cot²θ + 1 = cosec²θ	coth²x − 1 = cosech²x

Note the sign changes on the sinh² terms — this is Osborn's rule in action.

Double-Angle Formulae

Formula
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x
tanh 2x = 2tanh x / (1 + tanh²x)

Addition Formulae

Formula
sinh(A ± B) = sinh A cosh B ± cosh A sinh B
cosh(A ± B) = cosh A cosh B ± sinh A sinh B
tanh(A ± B) = (tanh A ± tanh B) / (1 ± tanh A tanh B)

Note: In cosh(A ± B), the sign on the right matches (unlike cos(A − B) = cos A cos B + sin A sin B, because Osborn's rule flips the product of two sines).

Derivatives

Function	Derivative
sinh x	cosh x
cosh x	sinh x
tanh x	sech²x
sech x	−sech x tanh x
cosech x	−cosech x coth x
coth x	−cosech²x

Note that d/dx(cosh x) = sinh x (no minus sign, unlike d/dx(cos x) = −sin x).

Integrals

Integral	Result
∫ sinh x dx	cosh x + C
∫ cosh x dx	sinh x + C
∫ tanh x dx	ln(cosh x) + C
∫ sech²x dx	tanh x + C
∫ cosech²x dx	−coth x + C
∫ sech x tanh x dx	−sech x + C

Solving Equations

To solve equations involving hyperbolic functions, convert to exponentials.

Example: Solve cosh x = 3.

• (eˣ + e⁻ˣ)/2 = 3
• eˣ + e⁻ˣ = 6
• Let u = eˣ: u + 1/u = 6 → u² − 6u + 1 = 0
• u = (6 ± √32)/2 = 3 ± 2√2
• x = ln(3 + 2√2) or x = ln(3 − 2√2)
• Since ln(3 − 2√2) = −ln(3 + 2√2), we get x = ±ln(3 + 2√2)

(This reflects the fact that cosh is even — symmetric solutions.)

Connection to Exponentials

Every result involving hyperbolic functions can be proved directly from the exponential definitions. This is the ultimate fallback:

• To prove an identity: substitute the exponential definitions and simplify
• To solve an equation: substitute and get a polynomial in eˣ

Exam Tips

• Osborn's rule is a shortcut, not a proof — you may need to verify from definitions
• The most-tested identity is cosh²x − sinh²x = 1 — use it to simplify expressions just as you would use cos² + sin² = 1
• d/dx(cosh x) = sinh x has no minus sign — the most common slip
• When solving cosh x = k, there are two solutions (±); when solving sinh x = k, there is one solution
• For integration, try rewriting in terms of sinh and cosh, then use the standard results above