## How to use the SECH function

**What is the SECH function?**

The SECH function calculates the hyperbolic secant of an angle.

**What is a secant?**

The trigonometric secant is a function that defines an angle of a right-angled triangle to the ratio of the hypotenuse to the adjacent side. It is also the inverse of the cosine, sec(Î¸) = 1/cos(Î¸).

Trigonometric functions are defined on the unit circle and deal with angles, circular geometry, however, hyperbolic functions are defined on the unit hyperbola and deal with hyperbolas, hyperbolic geometry.

**What is a hyperbolic secant?**

The hyperbolic secant is a trigonometric function defined as:

sech(x) = 1/cosh(x)

Where cosh(x) is the hyperbolic cosine function.

It is the reciprocal or inverse of the hyperbolic cosine function, just as the regular secant is 1/cos(x). The graph of y = sech(x) forms a S-shape, asymptotic to y = 0 as x approaches Â±infinity.

sech(x) is an even function, with sech(-x) = sech(x). It has a vertical asymptote at 0 and no horizontal asymptotes. Hyperbolic functions are analogs of trigonometric functions for hyperbolas rather than circles.

**What is a hyperbole?**

Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them called hyperbolas.

The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola.

The unit hyperbola looks like this:

x^{2}Â -y^{2}=1 and x^{2}Â -y^{2}=-1

These functions get closer, however, they never crosses or touches these dashed lines (asymptotes).

**What are the hyperbolic secant asymptotes?**

The hyperbolic secant function (sech(x)) has two vertical asymptotes located at x = 0 and x = Â±âˆž

This means that as x approaches 0 from either side, sech(x) approaches Â±âˆž and as x approaches positive or negative infinity, sech(x) approaches 0

**What are the main hyperbolic functions, their domain and range?**

Hyperbolic |
Domain |
Range |

sinh(x) | All real numbers | (-âˆž, âˆž) |

cosh(x) | All real numbers | [1, âˆž) |

tanh(x) | All real numbers | (-1, 1) |

sech(x) | All real numbers | (0, 1) |

**What is the hyperbolic domain?**

The domain of a hyperbolic function refers to the set of input values it is defined and valid for.

**What is the hyperbolic range?**

The range of a hyperbolic function refers to the set of possible output values it returns.

Hyperbolic Function |
Formula |
Asymptotes |

sinh | sinh x = (e^x - e^-x)/2 | y = e^x/2 and y = -e^-x/2 |

cosh | cosh x = (e^x + e^-x)/2 | y = 0 and y = âˆž |

tanh | tanh x = (e^x - e^-x)/(e^x + e^-x) | y = -1 and y = 1 |

coth | coth x = (e^x + e^-x)/(e^x - e^-x) | x = nÏ€ and y = -1 and y = 1 |

sech | sech x = 2/(e^x + e^-x) | y = 0 and y = âˆž |

csch | csch x = 2/(e^x - e^-x) | x = nÏ€ and y = 0 and y = âˆž |

Formula in cell C3:

### Excel Function Syntax

SECH(*number*)

### Arguments

number |
Â Required. An angle in radians. |

### Comments

Use the RADIANS function to convert degrees to radians.

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### Functions in 'Math and trigonometry' category

The SECH function function is one of many functions in the 'Math and trigonometry' category.

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