Trigonometry is about angles and triangles, there are relationships between the angles and the ratios of the lengths of side of the triangles. These ratios are represented by functions such as sine and cosine, which occur widely in mathematics and physics, even in fields which don't initially appear to be related to physical triangles.
On these pages we will consider two types of trigonometry:
 trigonometry based on a circle where: a²=b²+c²
 trigonometry based on a hyperbola where: a²=b²c²
For most cases we only need to consider traditional trigonometry based on a circle so feel free to ignore the hyperbolic trigonometry column. Hyperbolic trigonometry starts to become useful when we have a space with the Minkowski norm: (x²y²) the simplest case is the two dimensional space represented by double numbers as explained here.
Summary of results
This is a summary of the results in the above pages, for the derivation of these formulae see the links above.
Right angled triangle
circle based trigonometry  hyperbolic trigonometry 

Here we will use a unit circle, that is a circle where all the points are exactly 1 unit from the origin, which gives: b²+c²=1.  Here we will use a hyperbolic function where the minimum distance from the origin is 1 unit which gives: b²c²=±1. 
sine function sin(θ) = b 
hyperbolic sine function 
cosine function cos(θ) = c 
hyperbolic cosine function cosh(θ) = c 
tangent function tan(θ) = b / c 
hyperbolic tangent function tanh(θ) = d = b/c 
cosecant cosec(θ) = a / b 

secant sec(θ) = a / c 

cotangent cot(θ) = c / b 
Trig Identities
The values of sin, cos and tan are related to each other, for instance,
circle based trigonometry  hyperbolic trigonometry 

tan θ = sin θ / cos θ  tanh θ = sinh θ / cosh θ 
cos^{2}θ + sin^{2}θ = 1  cosh^{2}θ  sinh^{2}θ = 1 
cosec θ = 1 / sin θ  cosech θ = 1 / sinh θ 
sec θ = 1 / cos θ  sech θ = 1 / cosh θ 
cot θ = 1 / tan θ  coth θ = 1 / tanh θ 
hyperbolic
sinh(t) = (e^{t}  e^{t} )/2 = (e^{t}  1)/2*e^{t} = i sin(i t)
cosh(t) = (e^{t} + e^{t} )/2 = (e^{t} + 1)/2*e^{t} = cos(i t)
tanh(t) = (e^{t}  e^{t} )/(e^{t} + e^{t} ) = (e^{2t}  1)/(e^{2t} + 1)= i tan(i t)
asinh(x) = ln(x+√(x²+1))
acosh(x) = if (abs(x)<1) ln(x+√(1  x²))
else
ln(x+√(x²1))
atanh(x) = if (abs(x)<1) 0.5*ln((x1)/(x+1))
else 0.5*ln((x1)/(x+1))
Symmetric and Antisymmetric Properties
antisymmetric  sin(θ)= sin(θ)  sinh(θ)= sinh(θ)  sin(iθ)= i sinh(θ) 
symmetric  cos(θ)= cos(θ)  cosh(θ)= cosh(θ)  cos(iθ)= cosh(θ) 
antisymmetric  tan(θ)= tan(θ)  tanh(θ)= tanh(θ)  tan(iθ)= i tanh(θ) 
Complimentary and Supplementary Angles
 sin(θ  90°) = cos(θ)
 cos(θ  90°) = sin(θ)
 tan(θ  90°) = cot(θ)
NonRight Angled Triangles  The Sine Rule
a / sin A = b / sin B = c / sin C
NonRight Angled Triangles  The Cosine Rule
 cos A = (b^{2} + c^{2}  a^{2})/2*b*c
 cos B = (c^{2} + a ^{2}  b ^{2})/2*c*a
 cos C = (a^{2} + b ^{2}  c ^{2})/2*a*b
Area of triangles
In euclidean space the area of a triangle depends on length of 2 sides and one angle:
area = ^{1}/_{2} b * c*sin A
This is half the product of two sides and the sin of the included angle.
In hyperbolic space the area depends only on the internal angles:
π(α+β+γ)=CΔ
Addition and Subtraction formulae
circle based trigonometry  hyperbolic trigonometry 

sin(A+B) = sin A cos B + cos A sin B  sinh(A+B) = sinh A cosh B + cosh A sinh B 
cos(A+B) = cos A cos B  sin A sin B  cosh(A+B) = cosh A cosh B + sinh A sinh B 
tan(A+B) = (tan A + tan B)/ (1  tan A tan B) 
tanh(A+B) = (tanh A + tanh B)/ (1 + tanh A tanh B) 
sin(AB) = sin A cos B  cos A sin B  sinh(AB) = sinh A cosh B  cosh A sinh B 
cos(AB) = cos A cos B + sin A sin B  cosh(AB) = cosh A cosh B  sinh A sinh B 
tan(AB) = (tan A  tan B)/ (1 + tan A tan B) 
tanh(AB) = (tanh A  tanh B)/ (1  tanh A tanh B) 
Products of Trigonometric Functions
circle based trigonometry  hyperbolic trigonometry 

sin A sin B = 0.5(cos(AB)cos(A+B))  sinh A sinh B = 0.5(cosh(A+B)cosh(AB)) 
cos A cos B = 0.5(cos(AB)+cos(A+B))  cosh A cosh B = 0.5(cosh(A+B)+cosh(AB)) 
sin A cos B = 0.5(sin(AB)+sin(A+B))  sinh A cosh B = 0.5(sinh(A+B)+sinh(AB)) 
Double Angle and Half Angle Formulae
We can derive the double angle from products above where A=B. From this we can then derive the half angle formulae.
circle based trigonometry  hyperbolic trigonometry 

sin(2A) = 2 sin A cos A  sinh(2A) = 2 sinh A cosh A 
cos(2A) = cos^{2}A  sin^{2}A = 1  2*sin^{2}A  cosh(2A) = cosh^{2}A + sinh^{2}A 
tan(2A) = 2 tan A/(1  tan^{2}A)  tanh(2A) = 2 tanh A/(1 + tanh^{2}A) 
sin(t/2) =±√(0.5 (1 cos(t)))  sinh(t/2) =±√(0.5 (cosh(t)1)) 
cos(t/2) =±√(0.5 (1+ cos(t)))  cosh(t/2) =±√(0.5 (1+ cosh(t))) 
tan(t/2) = sin(t)/(1+cos(t))  tanh(t/2) = sinh(t)/(1+cosh(t)) 
derivation on this page.
Angles of Any Magnitude
 sin(0) = 0
 cos(0) = 1
 cos(90° + θ) = sin(θ)
 cos(90°  θ) = sin(θ)
 sin(180°  θ) = sin(θ)
 sin(180° + θ) = sin(θ)
 sec(180° + θ) = sec(θ)
 cosec(180°  θ) = cosec(θ)
 cos(270° + θ) = sin(θ)
 cos(270°  θ) =
 tan(270° + θ) = cotan(θ)
 tan(270°  θ) = cotan(θ)
 cot(360° + θ) = cot(θ)
 sin(360°  θ) = sin(θ)
Series
 sin(a) = a  ( a ^{3} / 3!) + ( a ^{5} / 5!)  ( a ^{7} / 7!) + ...
 cos(a) = 1  ( a ^{2} / 2!) + ( a ^{4} / 4!)  ( a ^{6} / 6!) + ...
 tan(a) = a + ( a ^{3} / 3) + ( 2a^{5} / 15) + ...
 sin^{1}(x) = x + ( x^{3} / 3!) + ( 9x^{5} / 5!) + (225 x^{7} / 7!) + ...
 cos^{1}(x) = PI/2  x  ( x^{3} / 3!)  (9 x^{5} / 5!)  ( 225 x^{7} / 7!) + ...
 tan^{1}(x) = x  ( x^{3} / 3) + ( x^{5} / 5)  ( x^{7} / 7) + ...
Other topics are as follows:
triangles  relationships between sides and angles in triangles 
trig functions  representing these relationships as continuous functions 
pythagoras  relationship between length of sides of a right angle triangle 
inverse trig functions  finding angle from ratio of sides arc sine, arc cosine, arc tan, etc. rectangular to polar conversion, need for atan2(y,x) function 
derived functions  relationship between trig functions double angle formula 
series  How to calculate the value of trig functions using an infinite series 
As a starting point goto this page: triangles