The sum formulae are given by
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b).
By taking
linear combinations of these, we obtain the
product formulae.
2sin(a)cos(b) = sin(a+b) + sin(a-b)
2cos(a)cos(b) = cos(a+b) + cos(a-b)
2sin(a)sin(b) = cos(a-b) - cos(a+b).
These results can be proved using a geometric argument, or by using the
exponential forms of
cosine and
sine, namely
2cos(x) = exp(ix) + exp(-ix)
2isin(x) = exp(ix) - exp(-ix).
They are immensely useful across most of
applied mathematics. One of the most simple consequences of them are the
double angle formulae, obtained by putting b = a:
sin(2a) = 2sin(a)cos(a)
cos(2a) = cos2(a) - sin2(a).
They can also be used to verify de Moivre's Theorem.