Secant function theory

The secant trig function can be expressed as the ratio of the sides of a right triangle. It is given by the ratio of hypotenuse to that of the adjacent side. It is denoted by secθ. Hence secθ = (length of hypotenuse) / (length of adjacent). It can also be expressed as the inverse of cosine function i.e. secθ = 1/cosθ. It is also referred as a circular function because secant of an angle can be expressed as ratio of (x, y) co-ordinates of circle with radius 1.

Properties & period of secant function

he figure drawn below represents the graph of secant function on vertical axis vs x. It is clearly evident from the graph that the period of secant function is equal to 2π.
secant function graph

Some of the important properties of secant function are given below to get a brief of the topic.

  • Secant function provides inverse results as that of cosine function.

  • Secant function is even just like that of cosine function i.e. sec(-x) = sec x.

  • The domain of secant function includes all the real numbers. The exception is (kπ + π/2, k) and (-1, 1).

  • Secant function has a range from (?, -1) to (1, +?).

  • sec x = cos(90-y). Both this functions are said to be co-functions of each other.

  • Secant function has a graph symmetrical in nature to y-axis and the period is 2π.

Lets have a table showing the values of secant function with respect to the change in secant angle. As shown below, value are given for θ = 0°, 30°, 45°, 60°, 90°.







sec θ





Not defined

Since cos 90° = 0, its inverse is an undefined value and hence sec 90° is not defined as shown. If we look at the table, it is clear that the values of secant for a particular angle is inverse to that of cosine. Moreover inverse secant functions can also be derived from the given values. An example given below is meant to provide you the required key information.

(1) Find sec(-60°):

Since secant is even in nature, sec(-60°) = sec 60° = 2.

Thus we have understood various properties related to secant including its domain and period.

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