A-Level Further Mathematics: Hyperbolic Functions

Hyperbolic functions, while seemingly abstract, are indispensable tools in advanced mathematics and physics, providing elegant solutions to problems involving growth, special relativity, and the shape of hanging cables. Mastering them extends your algebraic and calculus skills into new domains, bridging the gap between exponential and trigonometric thinking. This knowledge is not just academic; it underpins key models in engineering and theoretical physics.

Defining the Core Hyperbolic Functions

The three fundamental hyperbolic functions are defined using exponential functions. Unlike trigonometric functions, which are based on circles, hyperbolics are derived from hyperbolas, hence their name. Their definitions are precise and provide the easiest way to evaluate them.

The hyperbolic sine, denoted $sinhx$ (pronounced "sinch"), is defined as: $$sinhx=\frac{e^x-e^{-x}}{2}$$ The hyperbolic cosine, $coshx$ (pronounced "cosh"), is defined as: $$coshx=\frac{e^x+e^{-x}}{2}$$ From these, we define the hyperbolic tangent, $tanhx$ (pronounced "than"): $$tanhx=\frac{sinhx}{coshx}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

These definitions show that $coshx$ is an even function (symmetric about the y-axis), while $sinhx$ and $tanhx$ are odd functions. A simple way to remember them is that $coshx$ is the average of $e^x$ and $e^{-x}$, while $sinhx$ is half their difference. For example, to calculate $sinh(2)$, you would compute $(e^2-e^{-2})/2$, which approximates to 3.62686.

Fundamental Hyperbolic Identities and Their Proofs

Hyperbolic functions obey identities strikingly similar to, yet critically different from, trigonometric identities. These are not mere analogies; they are proven directly from the exponential definitions and are essential for simplifying expressions and solving equations.

The most crucial identity is the hyperbolic equivalent of $co{s}^2\theta +si{n}^2\theta =1$: $$cos{h}^{2}x-sin{h}^{2}x=1$$ Notice the minus sign, which is the key difference. To prove this, we square the definitions: $$cos{h}^{2}x=\frac{(e^x+e^{-x}{)}^2}{4}=\frac{e^{2x}+2+e^{-2x}}{4}$$ $$sin{h}^{2}x=\frac{(e^x-e^{-x}{)}^2}{4}=\frac{e^{2x}-2+e^{-2x}}{4}$$ Subtracting gives $(4)/4=1$.

Other important identities include:

• $sinh(2x)=2sinhxcoshx$
• $cosh(2x)=cos{h}^{2}x+sin{h}^{2}x=2cos{h}^{2}x-1=1+2sin{h}^{2}x$
• $cosh(A\pm B)=coshAcoshB\pm sinhAsinhB$
• $sinh(A\pm B)=sinhAcoshB\pm coshAsinhB$

You use these identities much like their trigonometric counterparts. For instance, to solve $2cos{h}^{2}x-3coshx+1=0$, you could let $y=coshx$, solve the quadratic $2y^2-3y+1=0$ to get $y=1$ or $y=0.5$, and then solve $coshx=1$ and $coshx=0.5$.

Differentiation and Integration of Hyperbolic Functions

The calculus of hyperbolic functions is remarkably straightforward, with derivatives that closely mirror—and are sometimes simpler than—those of trigonometric functions. This makes them easy to work with in solving differential equations.

The derivatives follow a cyclical pattern, but without sign changes:

• $\frac{d}{dx}(sinhx)=coshx$
• $\frac{d}{dx}(coshx)=sinhx$
• $\frac{d}{dx}(tanhx)=sec{h}^{2}x=1-tan{h}^{2}x$

These results are proven directly from differentiation of their exponential definitions. For example: $$\frac{d}{dx}(sinhx)=\frac{d}{dx}\left(\frac{e^x-e^{-x}}{2}\right)=\frac{e^x+e^{-x}}{2}=coshx$$

Reversing these gives the standard integrals, which you must commit to memory:

• $\int coshxdx=sinhx+c$
• $\int sinhxdx=coshx+c$
• $\int sec{h}^{2}xdx=tanhx+c$

A common exam question involves integration using a hyperbolic substitution, often leveraging the identity $cos{h}^{2}u-sin{h}^{2}u=1$. For example, to find $\int \sqrt{x^2+1}dx$, the substitution $x=sinhu$ transforms the integrand into $\sqrt{sin{h}^{2}u+1}=\sqrt{cos{h}^{2}u}=coshu$, greatly simplifying the problem.

Inverse Hyperbolic Functions and Logarithmic Forms

Just as we have inverse trigonometric functions, we have inverse hyperbolic functions: $arsinhx$ (or $sin{h}^{-1}x$), $arcoshx$, and $artanhx$. Their primary importance lies in their logarithmic forms, which provide explicit algebraic formulas for these inverses.

We derive the logarithmic form for $arsinhx$ by solving $x=sinhy=(e^y-e^{-y})/2$ for $y$. Multiplying by $2e^y$ gives $2x{e}^y=e^{2y}-1$, which is a quadratic in $e^y$: $(e^y{)}^2-2x(e^y)-1=0$. Using the quadratic formula and noting $e^y>0$, we get: $$arsinhx=\ln \left(x+\sqrt{x^2+1}\right)$$ Similar derivations yield: $$arcoshx=\ln \left(x+\sqrt{x^2-1}\right),x\ge 1$$ $$artanhx=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right),|x|<1$$

These forms are incredibly useful for integration. The derivative $\frac{d}{dx}(arsinhx)=\frac{1}{\sqrt{x^2+1}}$ leads directly to the standard result: $$\int \frac{1}{\sqrt{x^2+1}}dx=arsinhx+c=\ln \left(x+\sqrt{x^2+1}\right)+c$$ This is a more compact and elegant result than you might derive with trigonometric substitution.

Applications: Catenaries and Relativistic Velocity

Hyperbolic functions are not mere mathematical curiosities; they describe real physical phenomena with perfect accuracy. Two of the most compelling applications are the catenary curve and the mathematics of special relativity.

A catenary curve is the natural shape formed by a uniform flexible cable hanging under its own weight (e.g., power lines, suspension bridge cables). Its equation is $y=acosh(\frac{x}{a})$, where $a$ is a constant related to tension and density. The hyperbolic cosine is the solution to the differential equation governing the forces on the cable. This shape minimizes potential energy and provides uniform tension.

In Einstein's theory of special relativity, velocities do not add simply. If two spaceships are moving away from Earth at speeds $u$ and $v$ (as fractions of light speed), the relativistic velocity $w$ of one as seen from the other is given by: $$w=\frac{u+v}{1+uv}$$ This formula involves the hyperbolic tangent. Defining rapidity $\varphi =artanh(v)$, the addition law becomes the simple linear addition $w=tanh({\varphi }_u+{\varphi }_v)$. This demonstrates that rapidities, expressed via inverse hyperbolic tangents, are the additive quantities in relativistic frames, making hyperbolic functions the natural language for this domain.

Common Pitfalls

1. Confusing or Miscopying Identities: The most frequent error is forgetting the sign difference in the core identity, writing $cos{h}^{2}x+sin{h}^{2}x=1$ instead of $cos{h}^{2}x-sin{h}^{2}x=1$. Always derive it from the definitions if unsure. Similarly, mixing up the derivatives with trig derivatives (e.g., thinking $\frac{d}{dx}(coshx)=-sinhx$) is a trap.

1. Misapplying the Domain of Inverse Functions: The function $arcoshx$ is only defined for $x\ge 1$, and $artanhx$ only for $|x|<1$. Using values outside these domains, either in evaluation or integration, is invalid. For example, $\int \frac{1}{\sqrt{x^2-1}}dx=arcoshx+c$ only holds for $x>1$.

1. Integration Errors with Substitutions: When using a hyperbolic substitution like $x=asinhu$ to integrate an expression involving $\sqrt{a^2+x^2}$, a common mistake is to mishandle the differential ($dx=acoshudu$) or to incorrectly simplify $\sqrt{a^{2}sin{h}^{2}u+a^2}$. Always work through the substitution step-by-step: replace $x$ and $dx$, simplify the integrand using an identity, integrate, and then substitute back.

Summary

• Hyperbolic functions $sinhx$, $coshx$, and $tanhx$ are defined precisely by combinations of $e^x$ and $e^{-x}$, linking exponential and geometric ideas.
• They satisfy identities analogous to trigonometric ones, the most fundamental being $cos{h}^{2}x-sin{h}^{2}x=1$, which is proven directly from the definitions.
• Their calculus is straightforward: the derivative of $sinhx$ is $coshx$, and of $coshx$ is $sinhx$, leading to simple integral results.
• The inverse hyperbolic functions $arsinhx$, $arcoshx$, and $artanhx$ have explicit logarithmic forms which are essential for solving equations and evaluating certain integrals.
• These functions model real-world phenomena exactly, most notably the catenary curve $y=acosh(x/a)$ and the addition of velocities in Einstein's theory of special relativity.