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.

[画像: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 φ	Value cotφ
π/12	2 + √3
π/10	√(5 + 2 √5)
π/8	√2 + 1
π/6	√3
π/5	√(1 + 2 /√5)
π/4	1
3π/10	√(5 − 2 √5)
π/3	√3 /3
3π/8	√2 − 1
2π/5	√(1 − 2 /√5)
5π/12	2 − √3
π/2	0

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.

Function 1
√ or sqrt

Function 2
Γ or Gamma

Function 3
∏ or Prod

Function 4
∑ or Sum

Function 5
abs

Function 6
arccos

Function 7
arccot or arcctg

Function 8
arccsc or arccosec

Function 9
arcosh or arch

Function 10
arcoth or arcth

Function 11
arcsch

Function 12
arcsec

Function 13
arcsin

Function 14
arctan or arctg

Function 15
arg

Function 16
arsech or arsch

Function 17
arsinh or arsh

Function 18
artanh or arth

Function 19
ceil

Function 20
conj

Function 21
cos

Function 22
cosh or ch

Function 23
cot or ctg

Function 24
coth or cth

Function 25
csc or cosec

Function 26
csch

Function 27
exp

Function 28
fact

Function 29
floor

Function 30
I

Function 31
im

Function 32
J

Function 33
K

Function 34
ln

Function 35
log or lg

Function 36
^

Function 37
rand

Function 38
re

Function 39
sec

Function 40
sech

Function 41
sign

Function 42
sin

Function 43
sinh or sh

Function 44
tan or tg

Function 45
tanh or th

Function 46
Y

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