:::section[The Unit Hyperbola] Circular functions parametrize $X^2+Y^2=1$. Hyperbolic functions play the analogous role for the right-hand branch of $X^2-Y^2=1$. ::: :::theorem[Parametrization of the Unit Hyperbola] For every $t\in\mathbb{R}$, the point $$\begin{equation*} (\cosh t,\sinh t) \end{equation*}$$ lies on the unit hyperbola $X^2-Y^2=1$. ::: :::proof[Proof] Let $X=\cosh t$ and $Y=\sinh t$. Using the fundamental identity, $$\begin{equation*} X^2-Y^2=\cosh^2t-\sinh^2t=1. \end{equation*}$$ Moreover, $X=\cosh t\ge1$, so this parametrization traces the right-hand branch. $\square$ ::: :::remark[Hyperbolic and Circular Parametrizations] The point $(\cos t,\sin t)$ lies on the unit circle, while $(\cosh t,\sinh t)$ lies on the unit hyperbola. The names “circular” and “hyperbolic” record these geometric roles. ::: :::theorem[Monotonicity and Symmetry] The function $\sinh x$ is strictly increasing on $\mathbb{R}$. The function $\cosh x$ is even, strictly decreasing on $(-\infty,0]$, and strictly increasing on $[0,\infty)$. For every $x\in\mathbb{R}$, $$\begin{equation*} \cosh x-\sinh x=e^{-x}>0. \end{equation*}$$ ::: :::proof[Proof] Let $h>0$. By the addition identities, $$\begin{align*} \sinh(x+h)-\sinh x &=2\sinh\left(\frac h2\right) \cosh\left(x+\frac h2\right)>0. \end{align*}$$ Hence $\sinh$ is strictly increasing. For $0\le x<y$, $$\begin{align*} \cosh y-\cosh x &=2\sinh\left(\frac{x+y}{2}\right) \sinh\left(\frac{y-x}{2}\right)>0. \end{align*}$$ Thus $\cosh$ is strictly increasing on $[0,\infty)$. Its evenness gives the decreasing behavior on $(-\infty,0]$. The last inequality follows from the exponential identity above. $\square$ ::: :::key-formula[Domains and Ranges] $$\begin{align*} \operatorname{Dom}(\sinh)&=\mathbb{R}, &\operatorname{Ran}(\sinh)&=\mathbb{R},\\ \operatorname{Dom}(\cosh)&=\mathbb{R}, &\operatorname{Ran}(\cosh)&=[1,\infty),\\ \operatorname{Dom}(\tanh)&=\mathbb{R}, &\operatorname{Ran}(\tanh)&=(-1,1),\\ \operatorname{Dom}(\operatorname{sech})&=\mathbb{R}, &\operatorname{Ran}(\operatorname{sech})&=(0,1],\\ \operatorname{Dom}(\coth)&=\mathbb{R}\setminus\{0\}, &\operatorname{Ran}(\coth)&=(-\infty,-1)\cup(1,\infty),\\ \operatorname{Dom}(\operatorname{csch})&=\mathbb{R}\setminus\{0\}, &\operatorname{Ran}(\operatorname{csch})&=\mathbb{R}\setminus\{0\}. \end{align*}$$ ::: :::example[Reading a Range from an Identity] Since $\operatorname{sech}x=1/\cosh x$ and $\cosh x\ge1$, one has $$\begin{equation*} 0<\operatorname{sech}x\le1. \end{equation*}$$ The value $1$ occurs at $x=0$, while the value $0$ is approached but never attained. ::: :::exercise[Practice Questions] 1. Verify that $(\cosh(\ln2),\sinh(\ln2))$ lies on $X^2-Y^2=1$. 2. Explain why the range of $\tanh x$ excludes $\pm1$. 3. Determine all real $x$ for which $\cosh x=1$. 4. Use symmetry to describe the two branches of $\coth x$. ::: :::answer[Answers and Guidance] 1. The point is $(5/4,3/4)$, and $25/16-9/16=1$. 2. Since $$\begin{equation*} \tanh x=\frac{e^{2x}-1}{e^{2x}+1}, \end{equation*}$$ its value is strictly between $-1$ and $1$ for finite $x$. 3. Only $x=0$, because $\cosh x\ge1$ with equality only there. 4. $\coth$ is odd; one branch has values greater than $1$ for $x>0$, and the reflected branch has values less than $-1$ for $x<0$. ::: :::faq[Frequently Asked Questions] Q: Does $(\cosh t,\sinh t)$ cover both branches of the unit hyperbola? A: No. Since $\cosh t\ge1$, it covers the right-hand branch only. Q: Why is zero excluded from the range of $\operatorname{sech}$? A: A reciprocal of the finite positive number $\cosh x$ cannot be zero, although it tends to zero as $|x|$ grows. Q: Is $\cosh x>\sinh x$ still true when $x$ is negative? A: Yes. Their difference is $e^{-x}$, which is positive for every real $x$. ::: :::call-to-action[Continue Learning] subtitle: Use the exponential definitions and identities to derive the complete differentiation table. button: Differentiate Hyperbolic Functions|/real-analysis/functions/hyperbolic-and-inverse-hyperbolic-functions/differentiation-of-hyperbolic-functions button-ghost: Review This Section|/real-analysis/functions/hyperbolic-and-inverse-hyperbolic-functions :::