Secant function

The secant function that we are talking about is defined as one of the reciprocal of our basic three 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.


Secant function

In this mini-lesson, we will explore the world of the secant function. The journey will take us through secant function, unit circle, and angles, how to use them, and their various applications.

You can check out the interactive calculator to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. 

Lesson Plan


What Do You Mean By Secant Function?

Secant Definition

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.

Let us try to understand the concept of secant function by analyzing the four quadrants of the coordinate axis system.

Variation cycle of Sec x

Now, consider a unit circle centered at the origin of the coordinate plane.

Defining the T-ratios of theta

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) on 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:

\( \sec x  = \frac{{OP}}{{OQ}} \)

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


How to Plot a Graph for Secant Function?

The following figure shows a unit circle with center \(O\) at the origin, and a point \(P\) moving along the circumference of this circle.

The angle which \(OP\) makes with the positive direction of the \(x\)-axis is \(x\) (radians) 

Sec function - one point and one circle

\(PQ\) is the perpendicular dropped from \(P\) to the horizontal axis.

The secant function is the reciprocal of the cosine function.

\[\sec x= \frac{{1}}{cos x}\]

It is also considered as the secant formula.

We note that: \[\sec x = \frac{{OP}}{{OQ}} = \frac{{1}}{OQ}\]

As \(x\) varies, we study 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.

That is, finding \[\frac{{1}}{y}\] for each value of \(y\) on the curve \(\ y= \cos x\).

The table below shows some angles in radians:

\(x\) \(\ y= \cos x\). \[\sec x= \frac{{1}}{\cos x}\]
0 1 1
1 0.54 1.85
1.55 0.02 48.09
2 -0.42 -2.4
3 -0.99 -1.01
4 -0.65 -1.53

When the value of  \(\cos x\) is very small, the value of \(\sec x\) will become very large. 

First, we plot the graph for the cosine function.

Graph of cosine

Thus, we obtain the following graph of sec x:

graph of sec(x)

Now, on comparing both the graphs of cosine and secant function, we see that they are reciprocal of each other.

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.


Basic Properties of Secant Function

Properties of the secant function depend upon the quadrant in which the angle lies.

Variation cycle of Sec x

Sec function - one point and one circle

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.

This can also be observed for a few values of the sec function at different quadrants as shown in the table below:

Secant Degrees Secant Values
Secant 0° 0
Secant 30° 2/√3
Secant 45° √2
 Secant 60° 2
Secant 90° Not defined
Secant 120° -2
Secant 150° -2/√3
Secant 180° -1
Secant 270° Not defined
Secant 360° 1

What Do You Interpret From the Graph of Secant Function?

Range and Domain of Secant Function 

Graphing 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 figure above, the secant function is symmetric about the origin.

So the domain of secant is all real numbers except \(\frac {\pi }{2} + n\pi \). The range of secant is the set of all real numbers with a size greater than or equal to 1

Domain of secant: R except \(\frac {\pi }{2} + n\pi \)

Range of secant: (-∞,1] U [1, ∞)

Properties of Secant Functions

  • 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. 

Periodic graph of secant

  • So we can say that:

    \[\sec \left( {x + 2n\pi } \right) = \sec x\,\,\,{\rm{for \,\,\,every}}\,x\]

  • The secant function is an even function.
  • The secant function is an even function because \[ \sec(x) = \sec (-x). \]

 
important notes to remember
Important Notes
  • Sec function can be mathematically written as:\[Sec \theta = \dfrac{\text{Hypotenuse}}{\text{Base}}\]
  • The sec function has several distinct characteristics:
  1. They are periodic functions with a period of 2π.
  2. The domain of each function is R except \(\frac {\pi }{2} + n\pi \) and the range is (-∞,1] U [1, ∞).

Solved Examples

Example 1

 

 

Find the values of:

a) \(\sec {780^0}\)

b) \(\cos \left( { - {{660}^0}} \right)\)

Solution

(a) We have:

\[\sec \theta = \frac{{1}}{{cos \theta}} \\\implies \sec{780^0} = \dfrac{\text {1}} {\cos 780^0}\]

\[\cos{780^0} = cos ({60^0 + 2(360^0))= \cos {60^0} } \\\implies\sec \left( {{{780}^0}} \right) = \frac{1}{(\cos {{{60}^0)}}} = \frac{1}{({{{1/2})}}}= 2\]

\( \therefore 2\) 

(b) We note that:

\[\begin{align}- {660^0} &= - {720^0} + {60^0}\\ \Rightarrow  - {660^0} &\equiv {60^0}\\ \Rightarrow \cos \left( { - {{660}^0}} \right)  &= \cos \left( {{{60}^0}} \right)\\ &= \frac{1}{2}\end{align}\]

\( \therefore \frac{1}{2} \) 
Example 2

 

 

John wants to find the value of x for the given \(\triangle\)MNO. It is known that \( \sec \theta = 0.6\). Can you help him in solving this?

sec function

Solution

Given,   \[ \sec \theta = 0.6\]

Since,\[ \begin{align*} \sec \theta &=\dfrac{\text {Hypotenuse}}{\text{Base}} \\ \implies 0.6 &=  \dfrac{\text {Hypotenuse}}{\text{Base}} \\ \implies 0.6 &= \dfrac x5 \\ \implies x &= 3\end{align*}\]

\( \therefore\) x = 3
 

Thinking out of the box
Think Tank
  1. What can you say about the symmetry of the graph of the secant function?
  2. Is the secant function an odd function or an even function?

Interactive Questions

Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 

Let's Summarize

We hope you enjoyed learning about the secant function with the simulations and interactive questions. Now you will be able to easily solve problems on all secant functions.

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Frequently Asked Questions (FAQs)

1. How do you write a secant function?

A sine function can be written as \( \text {Sec x} = \dfrac {\text {Hypotenuse}}{\text {Base}} \) 

2. 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.

3. What is secant in terms of cosec?

\[Sec x = \frac{{1}}{cos x}\]

4. Where is the secant function undefined?

When the value of cosine is 0, the secant is undefined.

5. In what quadrant is secant negative?

Secant function is negative in the second and third quadrant.

6. What is the reciprocal of the secant function.

 Secant function is the reciprocal of the cosine function, 

7. What is the function of sec?

The sine function is equal to the ratio of the hypotenuse and the base of a right-angled triangle.

8. Which trigonometric functions are even?

Cosine and secant are the trigonometric functions that are even.

  
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