Trigonometry| Trigonometry |
Basic Plane Trigonometry > s.a. Hyperbolic
Functions.
* Simple relationships:
sin 
=
(exp{i}–exp{–i})
/ 2i , cos =
(exp{i}+exp{–i}
/ 2 , sin2 +
cos2 =
1 .
* Other algebraic relationships:
| sin 2 = 2 sin cos |
cos 2 = cos2 – sin2 |
sin /2 = [ (1 – cos )]1/2 |
cos /2 = [(1 +
cos )]1/2 |
| sin2 = (1 – cos
2) |
cos2 = (1
+ cos2) |
| sin cos =
sin
2 |
sin2 cos2 =
(1 – cos 4)
/ 8 |
| sin3 =
(3 sin – sin
3)/4 |
cos3 =
 (3
cos + cos 3) |
| sin3 cos =
(sin 2 – sin
4) |
sin4 =
( – 2
cos2 +
cos4) |
sin(   )
= sin cos cos sin |
cos( )
= cos cos  sin sin |
| sin sin =
[cos(–) – cos(+)] |
cos cos =
[cos(–)
+ cos(+)] |
| cos sin =
[sin(+) – sin(–)] |
sin cos =
(1 sin 2)1/2 |
* Integrals: If In = 
–Pi/2Pi/2 d
cosn ,
then I1 =
2, I2 =
/2, I3 =
4/3, I4 =
3/4.
References > s.a. simplex.
@ General: Maor 98 [history]; Andreescu & Feng 05 [problems].
@ 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 , and 
their
opposite angles,
then
sin /
sin a =
sin /
sin b = sin /
sin c .
* Law of cosines: Using the same notation,
cos a = cos b cos c + sin b sin
c cos .
* Triangle area: On a sphere of radius r, if ,
and are the internal angles of the triangle,
A = r2 ( +
+ – )
.
* Right triangles: If c is the length of the hypotenuse,
the three sides
satisfy cos c = cos a cos b; If is
the angle between c and a,
its sine and cosine can be expressed as
sin =
sin a / sin c = cos /
cos b , cos
= tan b /
tan c = cos a sin .
> Online resources: MathWorld page; University of Cambridge page; St Andrews University page.
Other Types
@ For different signatures/curvature: & Salgado; Herranz et al JPA(00)mp/99;
Catoni et al NCB(03)mp/05 [hyperbolic].
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23 dec 2007