Cotangent
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.
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:
- 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.
Trigonometric Identities Involving the Cotangent Function
Cotangent of Complementary, Supplementary and Conjugate Angles
- Cotangent of a Complementary Angle:
- Cotangent of a Supplementary Angle:
- Cotangent of a Conjugate Angle:
Cotangent of Negative Angles
- Cotangent of a Negative Angle:
Cotangent of Angles that Differs by 90º or 180º
- Cotangent of an Angle that Differs by 90º:
- Cotangent of an Angle that Differs by 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.
- 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.