Another set of identities worth knowing are the **factor formulae**. Since `sin(a+b) != sin(a) + sin(b)`

, these rules have been derived.

a + b a - b
sin(a) + sin(b) = 2sin ----- cos -----
2 2
a + b a - b
sin(a) - sin(b) = 2cos ----- sin -----
2 2
a + b a - b
cos(a) + cos(b) = 2cos ----- cos -----
2 2
a + b a - b
cos(a) - cos(b) = 2sin ----- sin -----
2 2

More likely than not, in a practical situation the equation *a*cos(θ) + *b*sin(θ)

will crop up. In such cases, the **harmonic form** is necessary.

The basic structure of the harmonic form is either `Rcos(θ ± α)`

or `Rsin(θ ± α)`

. (R > 0, α is acute). This is best explained with an example:

To express 3cosθ - 4 sinθ in the form Rcos(θ + α):
Let 3cosθ - 4sinθ ≡ Rcos(θ + α)
≡ R(cosθcosα - sinθsinα)
≡ Rcosθcosα - Rsinθsinα
Now equate the coefficients of cosθ and sinθ to obtain
3 = Rcosα (1) and 4 = Rsinα (2)
Squaring (1) and (2) and adding give:
R^{2}cos^{2}α + R^{2}sin^{2}α = 3^{2} + 4^{2}
R^{2}(cos^{2}α + sin^{2}α) = 25
∴ R^{2} = 25 (since cos^{2}α + sin^{2}α = 1)
∴ R = 5 (since R > 0)
Dividing (2) by (1) gives
Rsinα 4
----- = -
Rcosα 3
4
∴ tanα = -
3
∴ α = 53.1°
Therefore, we have
**3cosθ - 4sinθ = 5cos(θ + 53.1°)
**

The explanations in this section come from my own study. The specific example above comes from **Introducing Pure Mathematics**, *Smedley/Wiseman*.