Cotangent

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Drawing of the right triangle to calculate the secant

The cotangent is the reciprocal trigonometric ratio of the tangent. It is the reciprocal or multiplicative inverse of the tangent, that is, tan θ · cotθ = 1.

In a right triangle, the cotangent of the angle θ is defined as the ratio of the adjacent leg (b) to the opposite leg (a).

Cotangent formula

Like other trig functions, it is usually abbreviated. So, in a formula, the cotangent is abbreviated as cot (cotangent-, cotangens: from co ”mutually” + Latin tangens, that means “to touch” (Latin verb: tangere)).

Cotangent for Special Common Angles

The following table gives the values of cotangent for common angles:

Table of the cotangent of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)
Drawing on the goniometric circumference of the cotangent of the most characteristic angles and the sign of the secant in each quadrant

Properties of Cotangent

  • Domain: Cotangent domain (all real numbers), except n · π, where n is an integer. Or this casuistry: x ≠ ±π; ±2π; ±3π;… (that is, odd multiples of π).
  • Range: Cotangent range (all real numbers)
  • Symmetry: since cot (-x) = -cot (x) then cot (x) is an odd function and its graph is symmetric with respect to the origin (0, 0).
  • Increasing-decreasing behaviour: over one period and from 0 to 2π, sec (x) is decreasing.
  • End behaviour: The limits as x approaches π+ k · π do not exist since the function values oscillate between -∞ and +∞. Hence, there are no horizontal asymptotes. This is a periodic function with period π.
  • The derivative of cotangent function: The derivative of cotangent function
  • The integral of cotangent function: The integral of cotangent function

Graphical Representation of the Cotangent Function

Graph of the cotangent function

The cotangent is a periodic function with period π radians (180 degrees), so this section of the graph will be repeated in different periods.

Geometric Representation of the Cotangent

Drawing of the geometric representation of the cotangent

Relationship Between Cotangent and Other Trigonometric Functions

There are some basic trigonometric identities involving cotangent:

  • Relationship between cotangent and sine:
    Formula for the relationship of cotangent to sine
  • Relationship between cotangent and cosine:
    Formula for the relationship of cotangent to cosine
  • Relationship between cotangent and tangent:
    Formula for the relationship of cotangent to tangent
  • Relationship between cotangent and secant:
    Formula for the relationship of cotangent to secant
  • Relationship between cotangent and cosecant:
    Formula for the relationship of cotangent to cosecant

(1) Note: the sign depends on the quadrant of the original angle.

Trigonometric Identities Involving the Cotangent Function

Cotangent of Complementary, Supplementary and Conjugate Angles

Cotangent of Negative Angles

Cotangent of Angles that Differs by 90º or 180º

Other Reciprocal Trigonometric Ratios

Reciprocal trigonometric ratios are the multiplicative inverses of trigonometric ratios. These are:

  • Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1.
    Secant formula
  • Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.
    Cosecant formula

Reciprocal Trigonometric Ratios for Special Common Angles

The inverse trigonometric ratios for the most common angles (0º, 30º, 45º, 60º, 90º, 180º and 270º) are:

Table of inverse trigonometric ratios (cosecant, secant, cotangent) of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)

Relationships Between Trigonometric Functions

Any trigonometric ratio can be expressed in terms of any other. The following table shows the formulas with which each one is expressed as a function of the other:

Table of the relationship between trigonometric ratios

Note: the sign + or – depends on the quadrant of the original angle.

Trigonometric Ratios of Angle θ

Drawing of the right triangle to calculate the sine

If θ is one of the acute angles in a right triangle ABC, then:

  • The sine is the ratio between the length of the oppisite leg (a) divided by the length of the hypotenuse (c). In a formula, it is written simply as sin:
    Sine formula
  • The cosine is the ratio between the length of the adjacent leg (b) divided by the length of the hypotenuse (c). In a formula, it is written simply as cos:
    Cosine formula
  • The tangent is the ratio between the length of the opposite leg (a) divided by the length of the adjacent leg (b). In a formula, it is written simply as tan:
    Tangent formula

Trigonometric Functions

Trigonometric functions are also called circular functions. The reason is that in a right triangle that has been drawn on the coordinate axis, with the vertex of the angle θ at the center of a circle (O), that is, in standard position, its vertex B can go through all the points of this circle.

Drawing of the trigonometric functions of a triangle on a circle of radius 1

Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.


AUTHOR: Bernat Requena Serra

YEAR: 2021


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