Trigonometry| 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 |
* Derivatives:
d\(\,\tan^{-1}(x)\)/dx = \(1/(1+x^2)\).
* 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).
> Online resources:
see Wikipedia page.
Spherical Trigonometry
* History: Scholars do not agree
who invented spherical trigonometry; It might have been Hipparchos himself, or
possibly Menelaus two centuries later, or even Ptolemy a half century after that;
In any case, Ptolemy's Syntaxis mathematica contains a fully realized
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,
respectively, 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 .
@ References:
Van Brummelen PT(17)dec [history].
> Online resources: see
MathWorld page;
University of Cambridge page;
St-Andrews University page.
Other Types
@ Hyperbolic / Lorentzian: Catoni et al NCB(03)mp/05;
Dattoli & Del Franco a1002 [and special relativity];
in Dray 12;
Sorkin a1908 [and applications].
@ Parabolic: Dattoli et al a1102 [introduction].
@ For different signatures / curvature: & Salgado;
Herranz et al JPA(00)mp/99.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 1 dec 2019