The Art of Interface

# tan or tg — trigonometric tangent function

Category. Mathematics.

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

## 1. Definition

Tangent of the angle is ratio of the opposite leg to adjacent one.

## 2. Plot

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

Fig. 1. Plot of the tangent function y = tanx.

Function codomain is entire real axis.

## 3. Identities

Base:

sec2φ − tan2φ = 1

and its consequences:

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

By definition:

tanφ ≡ sinφ /cosφ ≡ 1 /cotφ

Properties — symmetry, periodicity, etc.:

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

Half-angle:

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

Double angle:

tan(2φ) = 2 tanφ /(1 − tan2φ)

Triple angle:

tan(3φ) = (3 tan2φ − tan3φ) /(1 − 3 tan2φ)

tan(4φ) = (4 tanφ − 4 tan3φ) /(1 − 6 tan2φ + tan4φ)

Power reduction:

tan2φ = [1 − cos(2φ)] /[1 + cos(2φ)]
tan3φ = [3 sinφ − sin(3φ)] /[3 cosφ + cos(3φ)]
tan4φ = [3 − 4 cos(2φ) + cos(4φ)] /[3 + 4 cos(2φ) + cos(4φ)]
tan5φ = [10 sinφ − 5 sin(3φ) + sin(5φ)] /[10 cosφ + 5 cos(3φ) + cos(5φ)]

Sum and difference of angles:

tan(φ + ψ) = (tanφ + tanψ) /(1 − tanφ tanψ)
tan(φψ) = (tanφ − tanψ) /(1 + tanφ tanψ)
tan(φ + ψ + χ) = (tanφ + tanψ + tanχ − tanφ tanψ tanχ) /(1 − tanφ tanψ − tanφ tanχ − tanψ tanχ)

Product:

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

Sum:

tanφ + tanψ = sin(φ + ψ) /(cosφ cosψ)
tanφ − tanψ = sin(φψ) /(cosφ cosψ)

Tangent of inverse functions:

tan(arctan x) ≡ x
tan(arcsin x) = x /√(1 − x2)
tan(arccos x) = √(1 − x2) /x

Some angles:

Angle φValue tanφ
00
π/122 − √3
π/10√(1 − 2 /√5)
π/8√2 − 1
π/6√3 /3
π/5√(5 − 2√5)
π/41
3π/10√(1 + 2 /√5)
π/3√3
3π/8√2 + 1
2π/5√(5 + 2√5)
5π/122 + √3
Table 1. Tangent for some angles.

## 4. Support

Trigonometric tangent function tan or tg of the real argument is supported by free version of the Librow calculator.

Trigonometric tangent function tan or tg of the complex argument is supported by professional version of the Librow calculator.

## 5. How to use

To calculate tangent of the number:

``tan(-1);``

To calculate tangent of the current result:

``tan(rslt);``

To calculate tangent of the angle φ in memory:

``tan(mem[φ]);``