Hyperbolic Functions — CIE A-Level Further Mathematics
1. Definitions and Graphs of Hyperbolic Functions ★★☆☆☆ ⏱ 15 min
Hyperbolic functions share many similarities with trigonometric functions, but have different parity and range properties:
- $\cosh x$ is even ($\cosh(-x) = \cosh x$), range $[1, \infty)$
- $\sinh x$ is odd ($\sinh(-x) = -\sinh x$), range $(-\infty, \infty)$
- $\tanh x$ is odd, range $(-1, 1)$
2. Hyperbolic Identities ★★☆☆☆ ⏱ 20 min
- $\sinh(A+B) = \sinh A \cosh B + \cosh A \sinh B$
- $\cosh(A+B) = \cosh A \cosh B + \sinh A \sinh B$
- $\cosh 2x = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1$
- $\sinh 2x = 2\sinh x \cosh x$
3. Inverse Hyperbolic Functions ★★★☆☆ ⏱ 20 min
Restricted domains make core hyperbolic functions one-to-one, so we can define inverse functions that can be written exactly in terms of natural logarithms.
4. Calculus of Hyperbolic Functions ★★★☆☆ ⏱ 25 min
Differentiating and integrating hyperbolic and inverse hyperbolic functions gives standard results that are frequently used for integrals involving quadratics under square roots.
Common Pitfalls
Why: Students often forget to flip the sign for terms containing a product of two sines
Why: The function is undefined for $x < 1$, so any result with this domain is invalid
Why: Confusion with derivative of $\tan x$ leads to incorrect negative sign
Why: When substituting $u = kx$, $du = kdx$ so $dx = \frac{du}{k}$, this factor is often missed