The Art of Interface

Article 11 — Appendix A.18

cot or ctg — trigonometric cotangent function

Category. Mathematics.

Abstract. Trigonometric cotangent: definition, plot, properties, identities and table of values for some angles.

Reference. This article is a part of Librow scientific formula calculator project.

1. Definition

Cotangent of the angle is ratio of the ajacent leg to opposite one.

2. Plot

Cotangent is π periodic function defined everywhere on real axis, except its singular points πn, where n = 0, ±1, ±2, ... — so, function domain is (πn, π(n + 1)), n∈N. Its plot is depicted below — fig. 1.

Fig. 1. Plot of the cotangent function y = cot x. Fig. 1. Plot of the cotangent function y = cotx.

Function codomain is entire real axis.

3. Identities

Base:

csc2φ − cot2φ = 1

and its consequences:

cotφ = ±√(1 − sin2φ) /sinφ
cotφ = ±cosφ /√(1 − cos2φ)
cotφ = ±1 /√(sec2φ − 1)
cotφ = ±√(csc2φ − 1)

By definition:

cotφ ≡ cosφ /sinφ ≡ 1 /tanφ

Properties — symmetry, periodicity, etc.:

cot−φ = −cotφ
cotφ = cot(φ + πn), where n = 0, ±1, ±2, ...
cotφ = −cot(π − φ)
cotφ = −tan(π + φ)
cotφ = tan(π/2 − φ)

Half-angle:

cot(φ/2) = ±√[(1 + cosφ) /(1 − cosφ)]
cot(φ/2) = sinφ /(1 − cosφ)
cot(φ/2) = (1 + cosφ) /sinφ
cot(φ/2) = cscφ + cotφ
cotφ = [1 − tan2(φ/2)] /[2 tan(φ/2)]

Double angle:

cot(2φ) = (cot2φ − 1) /(2 cotφ)

Triple angle:

cot(3φ) = (3 cot2φ − cot3φ) /(1 − 3 cot2φ)

Quadruple angle:

cot(4φ) = (1 + cot4φ − 6 cot2φ) /(4 cot3φ − 4 cotφ)

Power reduction:

cot2φ = [1 + cos(2φ)] /[1 − cos(2φ)]
cot3φ = [3 cosφ + cos(3φ)] /[3 sinφ − sin(3φ)]
cot4φ = [3 + 4 cos(2φ) + cos(4φ)] /[3 − 4 cos(2φ) + cos(4φ)]
cot5φ = [10 cosφ + 5 cos(3φ) + cos(5φ)] /[10 sinφ − 5 sin(3φ) + sin(5φ)]

Sum and difference of angles:

cot(φ + ψ) = (cotφ cotψ − 1) /(cotφ + cotψ)
cot(φ − ψ) = (cotφ cotψ + 1) /(cotψ − cotφ)
cot(φ + ψ + χ) = (cotφ + cotψ + cotχ − cotφ cotψ cotχ) /(1 − cotφ cotψ − cotφ cotχ − cotψ tanχ)

Product:

cotφ cotψ = [cos(φ − ψ) + cos(φ + ψ)] /[cos(φ − ψ) − cos(φ + ψ)]
tanφ cotψ = [sin(φ + ψ) + sin(φ − ψ)] /[sin(φ + ψ) − sin(φ − ψ)]

Sum:

cotφ + cotψ = sin(φ + ψ) /(sinφ sinψ)
cotφ − cotψ = sin(ψ − φ) /(sinφ sinψ)

Cotangent of inverse functions:

cot(arccot x) ≡ x
cot(arcsin x) = √(1 − x2) /x
cot(arccos x) = x /√(1 − x2)

Some angles:

Angle φValue cotφ
π/122 + √3
π/10√(5 + 2 √5)
π/8√2 + 1
π/6√3
π/5√(1 + 2 /√5)
π/41
3π/10√(5 − 2 √5)
π/3√3 /3
3π/8√2 − 1
2π/5√(1 − 2 /√5)
5π/122 − √3
π/20
Table 1. Cotangent for some angles.

4. Support

Trigonometric cotangent function cot or ctg of the real argument is supported by free version of the Librow calculator.

Trigonometric cotangent function cot or ctg of the complex argument is supported by professional version of the Librow calculator.

5. How to use

To calculate cotangent of the number:

cot(-1);

To calculate cotangent of the current result:

cot(rslt);

To calculate cotangent of the angle φ in memory:

cot(mem[φ]);