Trigonometric equations and their solution

  • J. E. Hebborn
  • C. Plumpton
Part of the Core Books in Advanced Mathematics book series (CBAM)


It is often useful to express a cos θ + b sin θ as a single term such as r cos (θγ), 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.


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Copyright information

© J. E. Hebborn and C. Plumpton 1984

Authors and Affiliations

  • J. E. Hebborn
    • 1
  • C. Plumpton
    • 2
  1. 1.University of London School Examinations DepartmentWestfield College, University of LondonUK
  2. 2.University of London School Examinations DepartmentQueen Mary College, University of LondonUK

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