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Secant Cosecant Cotangent

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08 September 2020                

Read time: 5 minutes

Introduction

Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a triangle. Trigonometry derives its name from Greek words trigon that means three angles and metron that means measure. The earliest known records of the subject are credited to the Egyptians and the Babylonians. However, Aryabhatta is said to have used the idea of sine function as early as 500 AD!

csc sec cot



What are Secant Cosecant Cotangent?

Trigonometric functions are defined as functions that in a right-angled triangle relate its angles to the ratio of the sides of the right-angled triangle. These are basically six in number. They are:

  • SINE
  • COSINE
  • TANGENT
  • COTANGENT 
  • SECANT
  • COSECANT

Of the 6 trig functions, the first three are fundamental and the other three are derived from the first three functions. We will discuss the three derived functions that are Secant Cosecant Cotangent in detail here. It is important to note here that the six basic trigonometric ratios are closed under reciprocals and complements of angles. 

However, first, let us briefly recapitulate the three fundamental trigonometric functions or trigonometric ratios.

csc sec cot triangle

 In the given right-angled triangle- ABC, we have AB as the side opposite to angle x; BC as the side adjacent to angle x; AC as the hypotenuse of the triangle. The trigonometric formulas are:

  • The sine of angle ∠x = opposite/hypotenuse = AB/AC
  • The cosine of angle ∠x = adjacent/hypotenuse  = BC/AC
  • The tangent of angle  ∠x = opposite/adjacent  = AC/BC

What is Cotangent?

The cotangent, also referred popularly as cot or cotan of ∠x is defined as the reciprocal of its tangent. Thus, cot x = 1/tan x = adjacent/opposite = BC/AB.

What is Cosecant?

Similarly, cosecant also referred to popularly as cosec or csc of ∠x is the reciprocal of the sine function. Thus, cosec x = 1/sin x =  hypotenuse/opposite = AC/AB.

What is Secant?

Finally, secant, popularly denoted as sec of ∠x, is defined as the reciprocal of its cosine function. Thus, sec x  = 1/cos x = hypotenuse/adjacent = AC/BC.


Basic Properties of Csc Sec Cot Functions

The values of trigonometric ratios and trigonometric functions are the same for various measures of angles ranging from 0  to 2π. They are presented here in a tabular form for your convenience.

 

 0

 π/6

 π/4

 π/3

 π/2

 π

 3π/2

 2π

 cosec 

 Not defined

 2

 √2

 2/3

 1

 Not defined

 -1

 Not defined

 sec

 1

 2/3

 2

 2

 Not defined

 -1

 Not defined

 1

 cot

 Not defined

 3

 1

 1/3

 0

 Not defined

 0

 Not defined

Sign of Trigonometric Functions

It is important to understand the signs of the basic trigonometric functions before proceeding to study their trigonometric graphs. Sign of cot, cosec, and sec are dependent on the signs of their parent functions,i.e., tan, sin, and cos in each quadrant. 

  • In the first quadrant, i.e. 0 to π/2, all three fundamental functions are positive. Thus, the derived functions are also positive in the quadrant.
  • In the second quadrant (π/2 to π ) however, only sin and cosec are positive, cos and tan are negative.
  • In the third quadrant, (π to 3π/2), sin and cos are negative, only tan and cot are positive.
  • The fourth quadrant (3π/2 to 2π) has only the cos and sec function as positive, sin, and tan stand negative.

Please note that though their values vary, the signs of cot, cosec, and sec functions remain the same as their parent functions in each quadrant. The following table will help you understand the sign convention of these trigonometric functions better.

 

 I         

 II

 III

 IV

 sin x

 +

 +

 -

 -

 cos x

 +

 -

 -

 +

 tan x

 +

 -

 +

 -

 cosec x

 + 

 +

 -

 -

 sec x

 +

 -

 -

 +

 cot x

 +

 -

 +

 -


Secant Graph, Cosecant Graph, Cotangent Graph

Before moving on to understanding the sec graph, cosec graph, and cot graph detail, let us have a look at them. In this section, we present to you the trigonometric graphs of the three derived trigonometric functions sec x, cosec x, cot x. We shall discuss the range, periodicity, and other properties of these functions elaborately in the next section. 

Please note that we plot the angles measured in radians on the X-axis and the function value at each angle is plotted on the Y-axis. Further, the signs of the functions in different quadrants correspond correctly in the graphical representation of the functions. 

The study of these graphs plays an important role in engineering and mechanical modeling, as well as mapping of different natural phenomena. 

Sec x graph

cosec x graph

cot x graph


Range Periodicity, and Other Properties


We know that sine and cosine functions are defined for all real numbers. Thus, we notice that for each real number x,
                                          -1 <= sin x <= 1 

                                          -1 <= cos x <= 1
Thus, sin x and cos x lie in the range that has an interval of [-1,1]. 
The domain of y = sin x and y = cos x is the set of real numbers.

Domain of Secant

As sec x is the reciprocal of cos x, its domain is the set
             {x: x ∈ R and x ≠ (2n+1)π/2, n ∈  Z}

Domain of Cosecant

As cosec x is the reciprocal of sin x, its domain is the set 
             {x: x ∈ R and x ≠ nπ, n ∈  Z} 

Domain of Cotangent

As cot x is the reciprocal of tan x, its domain is the set
             {x: x ∈ R and x ≠ nπ, n ∈  Z}

Range of Secant

Range of Secant is the set   {y: y ∈ R, y <= -1 or y >= 1}

Range of Cosecant

Range of cosecant is the set :-     {y : y ∈ R, y >= 1 or y <= -1}

Range of Cosecant

Range of Cotangent is the set of all real numbers.

Here, R denotes the set Real Numbers and Z denotes the set Integers. 

Further, the periodicity of the cot, cosec, and sec trigonometric functions can be mapped according to their base functions. 

Period of Secant

2π

Period of Cosecant

2π

Period of Cotangent

π


Summary

The six basic trigonometric functions are the foundation of trigonometry. The functions are of the same value as trigonometric ratios. Every angle between 0 and 360 has a specific value that it takes for all the six trigonometric functions. These values vary in sign and magnitude as per the nature of the function. Understanding the three fundamental functions makes it convenient to grasp the three derived functions that are cot, cosec, and sec. They can be derived from the three primary trigonometric functions. Their graphs come out to be smooth, periodic and can reveal a few properties about them.

Written by Aarti Agarwal, Cuemath Teacher


 

Frequently Asked Questions (FAQs)

Opposite of tan?

The reciprocal of tan is the cot function. It is defined as the ratio between adjacent and opposite sides of the given angle.

what is secant?

sec x = 1/cos x = hypotenuse/adjacent

sec full form?

Secant

Sec 0, Cosec 0, Cot 0?

 

 cosec 

 Not defined

 sec

 1

 cot      

 Not defined

Sec 30, Cosec 30, Cot 30?

 cosec 

 2

 sec

 2/3

 cot      

 √ 3  

Sec 45, Cosec 45, Cot 45?

 cosec 

 √2

 sec

 2

 cot

 1

Sec 60, Cosec 60, Cot 60?

 cosec 

 2/3

 sec

 2

 cot

 1/ 3

Sec 90, Cosec 90, Cot 90?

 cosec 

 1

 sec

 Not defined

 cot

 0

1/cosx?

sec x = 1/cosx = hypotenuse/adjacent

What is cotangent?

cotan or cot  x = cosx/sinx = adjacent/opposite


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