Harmonic Identities Rsin(x ± α), Rcos(x ± α)

The following identities, known as the harmonic identities, are very useful in solving certain types of trig. equation.

Try and learn them. However, it is a good exercise to try and prove them.

They are all very similar in the method of proof and are based on the addition identities.

I would encourage you to look at how I have proved one and then try the others as they are all very similar.

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 )

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

Show:

  1. A\sin x + B\cos x \equiv R\sin (x + \alpha )

Show:

  1. A\sin x - B\cos x \equiv R\sin (x - \alpha )

Show:

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

Show:

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




2018-08-08T17:54:15+00:00
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