The Art of Mathematics

Help 6.44

tan or tg — trigonometric tangent function

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.

Fgi. 1. Graph 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 ≡ 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)

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(φ − ψ) = (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(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:

tanφ + tanψ = sin(φ + ψ) /(cosφ cosψ)
tanφ − tanψ = sin(φ − ψ) /(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. Derivative and indefinite integral

Tangent derivative:

tan′x = sec2x ≡ 1 /cos2x

Indefinite integral of the tangent:

∫ tanx dx = −ln|cosx| + C

where C is an arbitrary constant.

5. How to use

To calculate tangent of the number:

tan(−1);

To get tangent of the complex number:

tan(−1+i);

To get tangent of the current result:

tan(rslt);

To get tangent of the angle φ in calculator memory:

tan(mem[φ]);

6. Support

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

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