Trigonometric equations and their solution

Abstract

It is often useful to express a cos θ + b sin θ as a single term such as r cos (θ − γ), where r is positive. This is possible if we can find r and y such that

$$\begin{gathered} r\cos \left( {\theta -\gamma } \right)\equiv r\left( {\cos \theta \cos \gamma +\sin \theta \sin \gamma } \right) \hfill \\ \quad \quad \quad \quad \;\;\;\equiv a\cos \theta +b\sin \theta . \hfill \\ \end{gathered}$$

Comparing the coefficients of cos θ and sin θ

$$\Rightarrow \quad r\cos \gamma =a,\quad \;r\sin \gamma =b.$$

(It can be shown that, if

$${{a}_{1}}\cos \theta +{{b}_{1}}\sin \theta \equiv {{a}_{2}}\cos \theta +{{b}_{2}}\sin \theta ,$$

where a₁, a₂, b₁, b₂ are constants, then a₁ = a₂, b₁ = b₂. This should be compared with equating coefficients in polynomials.) Dividing, we have tan γ = b/a and the situation shown in Fig. 3.1 for the case when a > 0, b > 0.