The application of trigonometric functions in the area of ​​mathematics and in other areas is widely used in our world. These functions are one of the basic mathematical functions in areas such as triangulation , which is used in criminal investigations and cell service . They are also used in navigation, surveying , computer graphics , and music theory . And one of the most important areas is the cotangent trigonometric function .

What is the cotangent?

The cotangent is equivalent to the result that is equal to the quotient between the adjacent leg and the opposite leg of an arc or an angle . It is the inverse function of the tangent.

  • Characteristics of the cotangent
  •  What is it for
  • How it is calculated
  • Formula
  • Derivative of the cotangent
  • Integral
  • Domain and range

Characteristics of the cotangent

The main characteristics of the cotangent are the following:

  • It is the inverse trigonometric ratio of the tangent.
  • It is the reciprocal or multiplicative inverse of the tangent, that is, so α · cot α = 1.
  • Its abbreviation is cot, cotg or cotan .
  • Its  domain is equal to R except a · π , where a is an integer.
  • Has a tour function is the co-domain of the cotangent, this is equal to R .
  • Its graph is a cotangentoid wave.

 What is it for

It is used in the same way that the sine, cosine and tangent functions are used . It can be used in function of a right triangle , using the opposite and adjacent sides of the triangle, or it can be used in function of the unit circle , which shows the angles in radians.

If you want to find the height of a particular tree based on the shadow it casts when the sun is at an angle of 30 degrees, we can find this information using cotangent using a 30-60-90 triangle.

It also serves to model real situations for physical phenomena that must describe waves such as sound , simple harmonic movement and others.

How it is calculated

It can be calculated as follows:

Cotangent = Adjacent leg / Opposite leg

Since it is the inverse function of the tangent, it can also be calculated by dividing 1 by the tangent.


The formula to find the cotangent is:

Cot α = 1 / tan α = contiguous leg / opposite leg = b / a

Derivative of the cotangent

The derivative of the cotangent of a given function is equal to the derivative of the function with negative sign divided by the sinus to the square of this function:

F (x) = cotg w

F´ (x) = (- w´) / (sin² w)

Applying the trigonometric rules, the derivative of the cotangent can also be defined as follows:

F´ (x) = – w´ · cosec² w

F´ (x) = – w´ · (1 + cotg² w)


The formula used to find the cotangent integral is as follows:

∫ csc x sec x dx

Domain and range

The tangent and the cotangent are related not only by the fact that they are reciprocal, but also by the behavior of their ranges . In reference to the coordinate plane, tangent is y / X, and cotangent is X / y.

The domains of these two functions are restricted , because on some occasions their relationships could have zeros in the denominator , but their ranges are infinite.

To find out the domain of the cotangent we must know its value and then proceed to divide:

cos (x) / sin (x)

It is important to mention that sin (x) should never take the value of the number 0.

The cotangent domain comprises the group of real numbers minus the multiples that π has so that K is a then an integer.

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