Secant Function

The secant function that we are talking about is defined as one of the reciprocal of our basic three trigonometric functions. So, we have cosecant which is the reciprocal of sine, secant which is the reciprocal of cosine, and cotangent is the reciprocal of the tangent function. The value of the secant function can be determined by taking the ratio of the hypotenuse and base of a right-angled triangle.

In this article, we will explore the concept of the secant function and understand its formula using the unit circle, and angles, how to use the formula, and their various applications and properties. We will also plot the secant function graph and determine its value at various angles.

The secant function is a periodic function in trigonometry. The secant function or sec function can be defined as the ratio of the length of the hypotenuse to that of the length of the base in a right-angled triangle. It is the reciprocal of cosine function and hence, is also written as sec x = 1 / cos x. Let us try to understand the concept of secant function by analyzing a unit circle centered at the origin of the coordinate plane.

A variable point P is taken on the circumference of the circle and it continues to move on the circumference of this circle. From the figure, we observe that P is in the first quadrant, and OP makes an acute angle of x radians with the positive x-axis. PQ is the perpendicular dropped from P (a point on the circumference) to the x-axis. The triangle is thus formed by joining the points O, P, and Q as shown in the figure, where OQ is the base, and PQ is the height of the triangle. Hence, the sec function for the above case can be mathematically written as:

sec x = OP / OQ

Here, x is the acute angle formed between the hypotenuse and the base of a right-angled triangle.

As discussed above, the formula for the secant function is given by the ratio of the hypotenuse and the base of a right-angled triangle. That is, we can write it mathematically as sec θ = Hypotenuse / Base. Also, we know that the secant function is the reciprocal of the cosine function. Therefore, we can also write its formula as sec θ = 1/ cos θ.

As we study the trigonometric table, we observe the values of the trigonometric functions at different angles. In this section, we will go through the value of the secant function for standard angles such as 0°, 30°, 45°, 60°, and 90° along with other angles like 180°, 270°, and 360° included. It is best to remember the values of the trigonometric ratios of these standard angles which help in various calculations.

As we observed the unit circle in the first section with center O at the origin, and a point P moving along the circumference of this circle. The angle that OP makes with the positive direction of the x-axis is x (radians). PQ is the perpendicular dropped from P to the horizontal axis. The secant function is the reciprocal of the cosine function, that is, sec x = 1 / cos x. It is also considered as the secant function formula. We note that: sec x = OP / OQ = 1 / OQ. As x varies, we note that the value of sec x varies with the variation of the length of OQ.

Graphing secant becomes very easy since we already know the cosine graph, so we can easily derive the graph for sec x by finding the reciprocal of each cosine value. When the value of cos x is very small, the value of sec x will become very large. That is, finding 1/y for each value of y on the curve y= cos x. The table below shows some angles in radians:

Also, we observe that whenever the value of the cosine function is zero, the secant function goes to infinity, which means when the value of cosine is 0 then, the secant is undefined. Thus, we obtain the following graph of sec x:

Looking at the secant function on a domain centered at the y-axis helps us bring out its symmetry. Thus, as we can see in the secant function graph above, the secant function is symmetric about the origin. So the domain of secant is all real numbers except for points (2n + 1)π/2. The range of secant is the set of all real numbers with a magnitude greater than or equal to 1. Thus, we have:

• Domain of secant function: R - (2n + 1)π/2
• Range of secant function: (-∞,-1] U [1, ∞)

Properties of the secant function depend upon the quadrant in which the angle lies. From the above secant graph and circle relation, we can see that value of the secant function is positive in the first and fourth quadrant whereas, in the second and third quadrant, it bears a negative value. Let us list some of the basic properties of the secant function:

• The secant function is a periodic function.

• A periodic function is a function which, when meets a specific horizontal shift, P, results in a function equal to the original function, i.e. f(x + P) = f(x), for all values of x within the domain of f. The secant graph repeats itself after 2π, which suggests the function is periodic with a period of 2π.

• So we can say that: sec(x+2nπ) = sec x, for every x

• The secant function is an even function because sec(-x) = sec x, for all x.
• Sec x has vertical asymptotes at all values of x = π/2 + nπ, n being an integer.
• The secant function graph is symmetric with respect to the y-axis.

Important Notes on Secant Function

• Sec function can be mathematically written as: Sec x = Hypotenuse / Base
• It is a periodic function with a period of 2π.
• The domain of the secant function is R - (2n + 1)π/2 and the range is (-∞,-1] U [1, ∞).

☛ Related Topics

• Cos Sec Tan
• Trigonometric Functions
• Slope of the Secant Line Formula

What is Secant Function in Trigonometry?

Secant function is one of the important trigonometric functions in trigonometry. It is equal to the ratio of the hypotenuse and the base of a right-angled triangle. Secant function is also known as the reciprocal of the cosine function.

How Do you Write a Secant Function?

We can write the formula for the secant function as sec x = Hypotenuse / Base. It can also be written as sec x = 1 / cos x as it is the reciprocal of cos x.

What is the Definition of Secant?

The secant function or sec function can be defined as the ratio of the length of the hypotenuse to that of the length of the base in a right-angled triangle.

What is Secant in terms of Cos?

Secant function in terms of cos is written as sec x = 1 / cos x.

Where is the Secant Function Not Defined?

The secant function is not defined at points where the value of the cosine function is equal to 0. The value of cos x is equal to zero at all real numbers except (2n + 1)π/2. So, the secant function is not defined at points (2n + 1)π/2, where n is an integer.

In What Quadrant is Secant Negative?

Secant function is negative in the second and third quadrant.

What is the Reciprocal of the Secant Function?

Using the reciprocal identities, we know that the reciprocal of the secant function is the cosine function.

Is Secant Function an Even Function?

Secant function is an even function because sec (-x) = sec x, for all x.

What is the Domain and Range of Secant Function?

The domain of the secant function is R - (2n + 1)π/2 and the range is (-∞,-1] U [1, ∞).

What is the Period of Secant Function?

The period of the secant function is equal to 2π.

Where are the Vertical Asymptotes on the Graph of the Secant Function?

The secant function has vertical asymptotes at points where the cosine function is equal to zero. Since sec x is the reciprocal of cos x, therefore sec x is not defined at points where cos x is 0. So, the secant function has vertical asymptotes at points (2n + 1)π/2, where n is an integer.