Article 10
Anti-hyperbolic Functions.

The equations $\frac{x}{a}=coshu,\frac{y}{b}=sinhu,\frac{t}{b}=tanhu$, etc., may also be expressed by the inverse notation $u={cosh}^{-1}\frac{x}{a},u={sinh}^{-1}\frac{y}{b},u={tanh}^{-1}\frac{t}{b}$, etc., which may be read: “$u$ is the sectorial measure whose hyperbolic cosine is the ratio $x$ to $a$,” etc.; or “$u$ is the anti-h-cosine of $\frac{x}{a}$,” etc.

Since there are two values of $u$, with opposite signs, that correspond to a given value of $coshu$, it follows that if $u$ be determined from the equation $coshu=m$, where $m$ is a given number greater than unity, $u$ is a two-valued function of $m$. The symbol ${cosh}^{-1}m$ will be used to denote the positive value of $u$ that satisfies the equation $coshu=m$. Similarly the symbol ${sech}^{-1}m$ in will stand for the positive value of $u$ that satisfies the equation $sechu=m$. The signs of the other functions ${sinh}^{-1}m,{tanh}^{-1}m,{coth}^{-1}m,{csch}^{-1}m$, are the same as the sign of $m$. Hence all of the anti-hyperbolic functions of real numbers are one-valued.

Prob. 14.

Prove the following relations:

$${cosh}^{-1}m={sinh}^{-1}\sqrt{m^2-1},{sinh}^{-1}m=\pm {cosh}^{-1}\sqrt{m^2+1},$$

the upper or lower sign being used according as $m$ is positive or negative. Modify these relations for ${sin}^{-1},{cos}^{-1}$.

Prob. 15.

In figure, Art. 1, let $OA=2,OB=1,AOB=60^{\circ }$; find the area of the hyperbolic sector $AOP$, and of the segment $AMP$, if the abscissa of $P$ is 3. (Find ${cosh}^{-1}$ from the tables for $cosh$.)