The addition formulae can be used to solve equations of the form:

\(a \cos{θ} + b \sin{θ} = c \)

Consider the expansion of \(R\sin{(θ + α)} \):

\(R\sin{(θ + α)} = R\sin{θ}\cos{α} + R\cos{θ}\sin{α} \)

By matching terms in the expansion with the equation, we form a pair of simultaneous equations:

\(R\sin{θ}\cos{α} = b \sin{θ} \implies R\cos{α} = b \)
\(R\cos{θ}\sin{α} = a \cos{θ} \implies R\sin{α} = a \)

Solving for \(R\) involves squaring both equations and adding to eliminate \(α\):

\(R^2\cos^2{α} + R^2\sin^2{α} = b^2 + a^2\)

\(\implies R^2\cancel{(\cos^2{α} + \sin^2{α})} = b^2 + a^2\)

\(\implies \boxed{R = \sqrt{b^2 + a^2}} \)

Solving for \(α\) involves dividing the equations and eliminating \(R\):

\(\dfrac{\cancel{R}\sin{α}}{\cancel{R}\cos{α}} = \dfrac{a}{b} \)

\(\implies \tan{α} = \dfrac{a}{b} \)

\(\implies \boxed{α = \arctan{\dfrac{a}{b}}} \)

The original equation \(a \cos{θ} + b \sin{θ} = c \) can therefore be rewritten as \(R\sin{(θ + α)} = c \) and solved the usual way.

This is not given in the formula book. You are not expected to memorise the result, but to be able to derive it.

Tip: Variations of this method will work for other forms of the addition formulae, i.e. \(R\sin{(θ-α)}\), \(R\cos{(θ+α)}\) and \(R\cos{(θ-α)}\). Questions will usually tell you which version of the addition formula to use (but not always).