Cosecant
The cosecant function is the reciprocal of the trigonometric function sine. Cosecant is one of the main six trigonometric functions and is abbreviated as csc x or cosec x, where x is the angle. In a rightangled triangle, cosecant is equal to the ratio of the hypotenuse and perpendicular. Since it is the reciprocal of sine, we write it as csc x = 1 / sin x.
In this article, we will explore the concept of cosecant function and understand its formula. We will plot the cosecant graph using its domain and range, explore the trigonometric identities of cosec x, its values, and properties. We will solve a few examples based on the concept of csc x to understand its applications better.
What Is a Cosecant Function?
Cosecant is the reciprocal of sine. We have six important trigonometric functions:
 Sine
 Cosine
 Tangent
 Cotangent
 Secant
 Cosecant
Since it is the reciprocal of sin x, it is defined as the ratio of the length of the hypotenuse and the length of the perpendicular of a rightangled triangle.
Consider a unit circle with points O as the center, P on the circumference, and Q inside the circle and join them as shown above. Since it is a unit circle, the length of OP is equal to the 1 unit. Consider the measure of angle POQ equal to x degrees. Then, using the cosecant definition, we have
csc x = OP/PQ
= 1/PQ
Cosecant Function Formula
Since the cosecant function is the reciprocal of the sine function, we can write its formula as
Cosec x = 1 / sin x
Also, since the formula for sin x is written as
Sin x = Perpendicular / Hypotenuse and csc x is the reciprocal of sin x, we can write the formula for the cosecant function as
Cosec x = Hypotenuse / Perpendicular
Domain and Range of Cosec x
As we discussed before, cosecant is the reciprocal of the sine function, that is, csc x = 1 / sin x, cosec x is defined for all real numbers except for values where sin x is equal to zero. We know that sin x is equal to for all integral multiples of pi, that is, sin x = 0 implies that that x = nπ, where n is an integer. So, cosec x is defined for all real numbers except nπ. Now, we know that the range of sin x is [1, 1] and csc x is the reciprocal of sin x, so the range of csc x is all real numbers except (1, 1). So the domain and range of cosecant are given by,
 Domain = R  nπ
 Range = (∞, 1] U [+1, +∞)
Cosecant Graph
Now that we know the domain and range of cosecant, let us now plot its graph. As we know cosec x is defined for all real numbers except for values where sin x is equal to zero. So, we have vertical asymptotes at points where csc x is not defined. Also, using the values of sin x, we have y = csc x as
 When x = 0, sin x = 0 and hence, csc x = not defined
 When x = π/6, sin x = ½, csc x = 2
 When x = π/4, sin x = 1/√2, csc x = √2
 When x = π/3, sin x = √3/2, csc x = 2/√3
 When x = π/2, sin x = 1, csc x = 1
So, by plotting the above points on a graph and joining them, we have the cosecant graph as follows:
Cosecant Identities
Let us now go through some of the important trigonometric identities of the cosecant function. We use these identities to simplify and solve various trigonometric problems.
 1 + cot²x = csc²x
 csc (π  x) = csc x
 csc (π/2  x) = sec x
 csc (x) = csc x
 csc x = 1 / sin x
 csc x = sec (π/2  x)
Properties of Cosecant Function
We have understood that the cosecant function is the reciprocal of the sine function and its formula. Let us now explore some of the important properties of the cosecant function to understand it better.
 The graph of cosec x is symmetrical about the xaxis.
 Cosecant Function is an odd function, that is, csc (x) = csc x
 The cosecant graph has no xintercepts, that is, the graph of cosecant does not intersect the xaxis at any point.
 The value of csc x is positive when sin x is positive and it is negative when sin x is negative.
 The period of csc x is 2π radians (360 degrees).
 Cosec x is not defined at the integral multiples of π.
Cosecant Values
To solve various trigonometric problems, we use the trigonometry table to memorize the values of the trigonometric functions which are most commonly used. The table given below shows the values of the cosecant function which help to simplify the problems and are easy to understand and remember.
X (radians) 
Csc x 

0 
Not defined 
π/6 
2 
π/4 
√2 
π/3 
2/√3 
π/2 
1 
3π/2 
1 
2π 
Not defined 
Important Notes on Cosecant Function
 Cosecant is the reciprocal of the sine function.
 It is equal to the ratio of hypotenuse and perpendicular of the right angles triangle.
 The cosecant graph has vertical asymptotes and has no xintercepts.
 Cosecant Function is defined at integer multiples of π.
ā Related Topics:
Cosecant Function Examples

Example 1: Find the values of the cosecant of angles A and C of triangle right angled at B, if AB = 12, AC = 13.
Solution: We know that csc x = Hypotenuse / Opposite Side.
Let us evaluate the value of BC first.
AC² = AB² + BC²
BC = √(AC²  AB²)
= √(13²  12²)
= √(169  144)
= √25
= 5 units
So, the values of cosecant of angles A and C are given by,
csc A = AC / BC
= 13/5
csc C = AC / AB
= 13/12
Answer: csc A = 13/5, csc C = 13/12

Example 2: Find the value of csc x if cot x = ¾ using cosecant identity.
Solution: To find the value of csc x, we will use the identity 1 + cot²x = csc²x
We have cot x = ¾
So,
1 + cot²x = csc²x
1 + (3/4)² = csc²x
1 + 9/16 = csc²x
csc²x = 25/16
csc x = √(25/16)
= 5/4
Answer: csc x = 5/4

Example 3: Find the value of cosecant of x if sin x = 4/13.
Solution: As we know that cosec x is the reciprocal of sin x, so we have
csc x = 1 / sin x
= 1 / (4/13)
= 13/4
Answer: csc x = 13/4
FAQs on Cosecant
What is Cosecant Function in Trigonometry?
The cosecant function is one of the important six trigonometric functions. It is the reciprocal of the sine function and hence, is equal to the ratio of Hypotenuse and Perpendicular of a rightangled triangle.
What is Cosecant Function Formula?
The cosecant function formula can be written in two different ways:
 csc x = 1/sin x
 csc x = Hypotenuse/Perpendicular OR Hypotenuse/Opposite Side
What is the Cosecant of an Angle?
The cosecant of an angle is equal to the ratio of the hypotenuse and opposite side of the angle in a rightangled triangle. We can also find the cosecant of angle using trigonometric identities.
What is the Difference between Secant and Cosecant?
Secant function is the reciprocal of the cosine function and the Cosecant function is the reciprocal of the sine function. Secant is the ratio of hypotenuse and adjacent side whereas cosecant is the ratio of the Hypotenuse and Opposite Side.
Is Csc the Inverse of Sin?
No, csc x is not the inverse of sin. It is the reciprocal of the sine function. The inverse of sin is called inverse sine or arcsin.
What is the Reciprocal of Cosecant?
The reciprocal of the cosecant function is the sine function. It is written as sin x = 1/csc x
What is the Period of Cosecant?
The values of the cosecant function repeat after every 2π radians, so the period of cosec x is equal to 2π radians (360 degrees).
Why is Cosecant the Reciprocal of Sine?
We know that sin x is the ratio of perpendicular and Hypotenuse of a rightangled triangle and Cosecant is the ratio of perpendicular and Hypotenuse, so cosecant is the reciprocal of sine. Also, the product of these two functions at an angle is always equal to one. Hence, cosecant is the reciprocal of the sine function.
Is Cosecant Function Graph Continuous?
Cosecant Graph is not continuous as it has vertical asymptotes at points where cosecant function is not defined. We know that cosec x is not defined at integer multiples of pi, so the cosecant function graph has a discontinuity at points nπ, where n is an integer.
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