The Art of Mathematics

Help 6.23

cot or ctg — trigonometric cotangent function

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 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(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. Derivative and indefinite integral

Cotangent derivative:

cot′x = −csc2x ≡ −1 /sin2x

Indefinite integral of the cotangent:

∫ cotx dx = ln|sinx| + C

where C is an arbitrary constant.

5. How to use

To calculate cotangent of the number:

cot(−1);

To get cotangent of the complex number:

cot(−1+i);

To get cotangent of the current result:

cot(rslt);

To get cotangent of the angle φ in calculator memory:

cot(mem[φ]);

6. Support

Trigonometric cotangent of the real argument is supported in free version of the Librow calculator.

Trigonometric cotangent of the complex argument is supported in professional version of the Librow calculator.