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.

Cosecant is the reciprocal of sine. We have six important trigonometric functions:

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

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

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,

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

So, by plotting the above points on a graph and joining them, we have the cosecant graph as follows:

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.

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.

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

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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

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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 right-angled triangle.

The cosecant function formula can be written in two different ways:

The cosecant of an angle is equal to the ratio of the hypotenuse and opposite side of the angle in a right-angled triangle. We can also find the cosecant of angle using trigonometric identities.

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.

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.

The reciprocal of the cosecant function is the sine function. It is written as sin x = 1/csc x

The values of the cosecant function repeat after every 2π radians, so the period of cosec x is equal to 2π radians (360 degrees).

We know that sin x is the ratio of perpendicular and Hypotenuse of a right-angled 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.

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.