The sech() Command

Command Summary

Takes the hyperbolic secant of a number.

Command Syntax

sech(value)

• Press 2nd MATH to enter the MATH menu.
• Press C to enter the Hyperbolic submenu.
• Press 5 to select sech(.

This command requires a calculator with AMS version 2.07 or higher (it will also work on any TI-89 Titanium or Voyage 200 calculator)

1 byte

The sech() command returns the hyperbolic secant of a number. Along with 11 other trig and hyperbolic functions, it was added in AMS version 2.07; on earlier versions, sech(x) can be replaced by 1/cosh(x).

As long as the calculator is in radian mode, sech() can be used with complex numbers according to the rule that sech(ix)=sec(x) and sec(ix)=sech(x). This rule only works in radian mode, and sech() of a complex number will return a domain error when working in degrees or gradians.

Occasionally, sech() can compute an exact result; most of the time, the calculator will leave an expression with sech() alone unless it's in approximate mode (or you force an approximation). When sech() is used with symbolic expressions, the calculator can go back and forth between the sech() expression and its exponential equivalent.

:sech(0)
1
:expand(sech(x))
2*e^x/((e^x)^2+1)
:comDenom(2*e^x/((e^x)^2+1))
1/cosh(x)


If sech() is applied to a list, it will take the hyperbolic secant of every element in the list. However, it can't be applied to matrices the way cosh() can (this is probably an oversight; all the trig and hyperbolic functions that were present in all AMS versions work with matrices, but the ones added in version 2.07 do not).

# Formulas

The definition of hyperbolic secant is, by analogy with sec(), the reciprocal of cosh():

(1)
\begin{align} \operatorname{sech}{x}=\frac{1}{\cosh{x}} = \frac{2}{e^x+e^{-x}} \end{align}

# Error Conditions

260 - Domain error happens when taking sech() of a complex number in degree or gradian mode.