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Article 11 — Appendix A.18

cot or ctg — trigonometric cotangent function

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

1. Definition

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

2. Graph

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 graph is depicted below — fig. 1.

Fig. 1. Graph of the cotangent function y = cot x Fig. 1. Graph 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(φψ) = (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(φψ) + cos(φ + ψ)] /[cos(φψ) − cos(φ + ψ)]
tanφ cotψ = [sin(φ + ψ) + sin(φψ)] /[sin(φ + ψ) − sin(φψ)]

Sum:

cotφ + cotψ = sin(φ + ψ) /(sinφ sinψ)
cotφ − cotψ = sin(ψφ) /(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 is supported in:

Trigonometric cotangent function of the complex argument cot or ctg is supported in:

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]);