Trigonometry

sin (a + b) = sin a cos b + cos a sin b .

(A.4)

sin (2 a) = 2 sin a cos a .

(A.5)

cos (a + b) = cos a cos b - sin a sin b .

(A.6)

cos (2 a) = {cos}^{2} a - {sin}^{2} a .

(A.7)

From A.7 and ${cos}^{2} a + {sin}^{2} a = 1$ :

{cos}^{2} a = \frac{1}{2} (1 + cos (2 a))

(A.8)

and

{sin}^{2} a = \frac{1}{2} (1 - cos (2 a)) .

(A.9)

The following are some of the so-called product to sum identities:

cos (a) cos (b) = \frac{1}{2} [cos (a - b) + cos (a + b)] .

(A.10)

sin (a) sin (b) = \frac{1}{2} [cos (a - b) - cos (a + b)] .

(A.11)

sin (a) cos (b) = \frac{1}{2} [sin (a + b) + sin (a - b)],

(A.12)

where $a$ is the argument of the sine in Eq. (A.12).