Math Doubts

Secant

A term that represents a ratio of lengths of hypotenuse to adjacent side at an angle of a right triangle is called the secant.

Introduction

Secant is a name and it is introduced in trigonometry to represent the ratio of lengths of hypotenuse to adjacent side at a particular angle in a right angled triangle. It is usually written in ratio form and alternatively as secant with angle.

1. Ratio

The value of secant at an angle is calculated by the ratio of lengths of hypotenuse to adjacent side.

$\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}$

Hence, secant is called generally as a trigonometric ratio.

2. Function

Mathematically, the value of secant at an angle is written in alternate way by just writing secant in its short form $\sec$ and then respective angle of the right triangle.

For example, if angle of a right triangle is represented by $x$, then secant of angle $x$ is written as $\sec{x}$ in trigonometry and the $\sec{x}$ is a function form. Hence, it is generally called as secant function in mathematics.

Thus, sec functions like $\sec{A}$, $\sec{\alpha}$, $\sec{\beta}$, and etc. are defined in trigonometry.

Mathematical form

$\Delta BAC$ is a right triangle and angle of this triangle is denoted by theta ($\theta$).

right angled triangle

secant of angle is written as $\sec{\theta}$ in this case.

$\sec{\theta} \,=\, \dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}$

It is actually used as a formula to calculate the value of secant at any angle of the right triangle.

In this example, $AC$ is length of hypotenuse and $AB$ is length of adjacent side (base).

$\,\,\, \therefore \,\,\,\,\,\, \sec{\theta} \,=\, \dfrac{AC}{AB}$

Values

The list of exact values of secant functions in fraction and decimal forms in a table with proofs.

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