The cosecant function is an old function in the area of mathematics . It was mentioned in the works of GJ von Lauchen Rheticus in 1596 and E. Gunter around 1620. It was widely used by L. Euler in 1748 and T. Olivier, Wait and Jones in 1881. The classical definition of function cosecant for real arguments is: “the cosecant of an angle in a right angle triangle is the relationship between the length of the hypotenuse and the length of the opposite angle “. This description is valid for when this triangle is not degenerate. This approximation can be expanded to arbitrary real values if the arbitrary point in the Cartesian plane is considered and defined as the ratio assuming that α is the value of the angle between the positive direction of the axis and the direction from the origin to the point.


What is the cosecant?

The inverse function of the sine of an arc or of an angle is known as a cosecant . It is a term widely used in the area of ​​mathematics when it is related to the area of geometry .

  • Characteristics of the cosecant
  • What is it for
  • How it is calculated
  • Derivative of cosecant
  • Integral
  • Domain and range
  • Inverse function

Characteristics of the cosecant

The main characteristics of the cosecant are the following:

  • It is represented by the letters csc or cosec .
  • It is an inverse trigonometric relationship to the sine .
  • It is known as the multiplicative inverse .
  • Its domain is R – {k · π} with k  ∈ Z
  • The path of the cosecant is R – (- 1, 1).
  • It does not cut the X axis, nor does it cut the Y axis.
  • An important characteristic is that it is odd, which means that it is symmetric with respect to the origin : cosec (- x) = – cosec (x).
  • It also has infinite minima that are relative at points of the form (π / 2 + 2 k π, 1) – with k ∈ Z
  • It is periodic of period 2π cosec (x) = cosec (x + 2π).
  • It also has vertical asymptotes at points of the form x = k π with k∈
  • It is not limited .

What is it for

Like the other trigonometric functions, they are useful, for example, in the field of astronomy since it is used to measure distances between stars, in measuring distances between geographical points, and in satellite navigation systems. They are very functional in the case of the construction of houses or buildings to know the measures that must be done. It is also useful in civil engineering, to calculate distances, angles of inclination or superelevation of a road. In scientific development it is used in the elaboration of numerical methods by mathematicians to make differential equation or solve an integral that cannot be worked with conventional methods.

How it is calculated

The formula to find the cosecant is as follows:

Csc α = 1 / sin α = c / a 

Derivative of cosecant

The derivative of the cosecant of a function will be the same unless the cosecant of the function times the cotangent of the function and also the derivative of the function. The formula is as follows:

F (x) = cosec u

F ‘(x) = – (u ㆍ cosec u)  / (sen² u) = – ㆍ cosec or ㆍ cotg or


The formula used for the integral of the cosecant is as follows:

∫ csc u ㆍ du = – log (cot u + csc u)

This cosecant su can find out when derivation formulas are mastered

Domain and range

It is important to remember that the domain of a function is made up of all the input values ​​that a trigonometric function can carry out. The cosecant domains are restricted and can only be used for the functions of measuring the angles with the output numbers. Every time then that the terminal side that has an angle is along the X-axis (where y is equal to 0), the cosecant function of the angle cannot be executed .

Inverse function

In the area of ​​trigonometry, the inverse function of the cosecant is known as the arcoscant of an angle. This property is symbolized as:  arccosec α or it can also be identified as arccsc α.  Its meaning in geometry is the angle whose cosecant is alpha.

Y = arccsc (x)

X = csc (y)

From this definition we can deduce some important expressions, such as:

Arccosec (-x) = – arccosec x

Arccsc (x) = arcsin (1 / x)

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