Trigonometry

Basic Plane Trigonometry > s.a. Hyperbolic Functions.
* History: The subject evolved during the third century BC, but Hipparchos (190 BC – 120 BC) is usually considered the "father of trigonometry".
* Simple relationships:

sin α = (exp{iα}–exp{–iα}) / 2i ,   cos α = (exp{iα}+exp{–iα} / 2 ,   sin2α + cos2α = 1 .

* Other algebraic relationships:

 sin 2α = 2 sin α cos α cos 2α = cos2α – sin2α sin α/2 = [$$1\over2$$(1 – cos α)]1/2 cos α/2 = [$$1\over2$$(1 + cos α)]1/2 sin2α = $$1\over2$$(1 – cos 2α) cos2α = $$1\over2$$(1 + cos 2α) sin α cos α = $$1\over2$$sin 2α sin2α cos2α = (1 – cos 4α) / 8 sin3α = (3 sin α – sin 3α)/4 cos3α = $$1\over4$$(3 cos α + cos 3α) sin3α cosα = $$1\over4$$(sin 2α – $$1\over2$$sin 4α) sin4α = $$1\over4$$($$3\over2$$ – 2 cos 2α + $$1\over2$$cos 4α) sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β $$\mp$$ sin α sin β sin α sin β = $$1\over2$$[cos(α–β) – cos(α+β)] cos α cos β = $$1\over2$$[cos(α–β) + cos(α+β)] cos α sin β = $$1\over2$$[sin(α+β) – sin(α–β)] sin α ± cos α = (1 ± sin 2α)1/2

* Integrals: If In = –π/2π/2 dθ cosn θ, then I1 = 2, I2 = π/2, I3 = 4/3, I4 = 3π/4.

References > s.a. Products [infinite trigonometric products]; rotations [including imaginary angles]; simplex.
@ General: Maor 98 [history]; Andreescu & Feng 05 [problems]; Adrian 07 [WS; encyclopedia of proofs and games].
@ Identities: Gervois & Mehta JMP(95), JMP(96).

Spherical Trigonometry
* Law of sines: If a, b and c are the three sides of a triangle on the surface of a unit sphere (whose values are the angles subtended at the center), and α, β and γ their opposite angles, then

sin α / sin a = sin β / sin b = sin γ / sin c .

* Law of cosines: Using the same notation,

cos a = cos b cos c + sin b sin c cos α .

* Triangle area: On a sphere of radius r, if α, β and γ are the internal angles of the triangle,

A = r2 (α + β + γ – π) .

* Right triangles: If c is the length of the hypotenuse, the three sides satisfy cos c = cos a cos b; If α is the angle between c and a, its sine and cosine can be expressed as

sin α = sin a / sin c = cos β / cos b ,   cos α = tan b / tan c = cos a sin β .

> Online resources: see MathWorld page; University of Cambridge page; St-Andrews University page.

Other Types
@ Hyperbolic: Catoni et al NCB(03)mp/05; Dattoli & Del Franco a1002 [and special relativity]; in Dray 12.
@ Parabolic: Dattoli et al a1102 [introduction].
@ For different signatures / curvature: & Salgado; Herranz et al JPA(00)mp/99.