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.
Cotangent of the angle is ratio of the ajacent leg to opposite one.
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.
[画像: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.
Base:
csc2φ − cot2φ = 1and its consequences:
cotφ = ±√(1 − sin2φ) /sinφBy definition:
cotφ ≡ cosφ /sinφ ≡ 1 /tanφProperties — symmetry, periodicity, etc.:
cot−φ = −cotφHalf-angle:
cot(φ/2) = ±√[(1 + cosφ) /(1 − cosφ)]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)]Sum and difference of angles:
cot(φ + ψ) = (cotφ cotψ − 1) /(cotφ + cotψ)Product:
cotφ cotψ = [cos(φ − ψ) + cos(φ + ψ)] /[cos(φ − ψ) − cos(φ + ψ)]Sum:
cotφ + cotψ = sin(φ + ψ) /(sinφ sinψ)Cotangent of inverse functions:
cot(arccot x) ≡ xSome 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 |
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.
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]);