Formula, Graph, Domain, Range | Cosecant Function - Cuemath

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 right-angled 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.

1.	What Is a Cosecant Function?
2.	Cosecant Function Formula
3.	Domain and Range of Cosec x
4.	Cosecant Graph
5.	Cosecant Identities
6.	Properties of Cosecant Function
7.	Cosecant Values
8.	FAQs on Cosecant Function

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 right-angled 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 x-axis.
• Cosecant Function is an odd function, that is, csc (-x) = -csc x
• The cosecant graph has no x-intercepts, that is, the graph of cosecant does not intersect the x-axis 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 x-intercepts.
• Cosecant Function is defined at integer multiples of π.

☛ Related Topics:

• Cosine
• Trigonometric Table
• Trigonometric Ratios