Equations using harmonic identities

The Harmonic Identities are useful when solving equations with the following forms.

A sin x ± B cos x = C or A cos x ± B sin x = C

The Harmonic Identities:

  • A\sin x + B\cos x \equiv R\sin (x + \alpha )
  • A\sin x - B\cos x \equiv R\sin (x - \alpha )
  • A\cos x + B\sin x \equiv R\cos (x - \alpha )
  • A\cos x - B\sin x \equiv R\cos (x + \alpha )

    \[ \text{ where  \ \ }R>0 \text{ and } R = \sqrt {{A^2} + {B^2}} \text{ \ \ and \ \ } \alpha  = {\tan ^{ - 1}} \dfrac{B}{A}\]

In the examples that follow, I show you how using the harmonic identities helps solve these types of equations.

Solve:

  1. 2\sin \theta  - 3\cos \theta  = 1 \hphantom{aa}\text{for}\hphantom{aa}{0^ \circ } \le \theta  \le {360^ \circ }

Solve:

  1. 3\cos \theta  - 4\sin \theta  = 2 \hphantom{aa}\text{for}\hphantom{aa} 0^ \circ \le \theta  \le 360^ \circ




2018-08-08T17:55:17+00:00
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