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Article 11 — Appendix A.30

tan or tg — trigonometric tangent function

Category. Mathematics.

Abstract. Trigonometric tangent: 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

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

2. Graph

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 graph is depicted below — fig. 1.

Fig. 1. Graph of the tangent function y = tan x Fig. 1. Graph 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 ≡ /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)

Quadruple angle:

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 is supported in:

Trigonometric tangent function of the complex argument tan or tg is supported in:

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]);