Cartesian Coordinate Systems - 3D Math Primer for Graphics and Game Development - page 26

Figure 1.20
The Pythagorean theorem

The following identities involve taking a trig function on the sum or
difference of two angles:

sin(a + b) = sinacosb + cosasinb,
sin(a − b) = sinacosb − cosasinb,

Sum and difference
identities

(1.1)

cos(a + b) = cosacosb − sinasinb,
cos(a − b) = cosacosb + sinasinb,

tana + tanb
1 − tanatanb ,

tan(a + b) =

tana − tanb
1 + tanatanb .

tan(a − b) =

If we apply the sum identities to the special case where a and b are the
same, we get the following double angle identities:

sin 2θ = 2 sinθ cosθ,
cos 2θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ,

Double angle identities

2 tanθ
1 − tan 2 θ .

tan 2θ =

We often need to solve for an unknown side length or angle in a triangle,
in terms of the known side lengths or angles. For these types of problems
the law of sines and law of cosines are helpful. The formula to use will
depend on which values are known and which value is unknown. Figure 1.21
illustrates the notation and shows that these identities hold for any triangle,
not just right triangles:

sinA
a

= sinB
b

= sinC
c

,

Law of sines