# Cotangent

(No Ratings Yet)

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).

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:

## Properties of 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: (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 integral of cotangent function:

## Graphical Representation 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.

## Relationship Between Cotangent and Other Trigonometric Functions

There are some basic trigonometric identities involving cotangent:

• Relationship between cotangent and sine:
• Relationship between cotangent and cosine:
• Relationship between cotangent and tangent:
• Relationship between cotangent and secant:
• Relationship between cotangent and cosecant:

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

## 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.
• Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.

## 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:

## 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:

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

## Trigonometric Ratios of Angle θ

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:
• 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:
• 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:

## 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.

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