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

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

Show:

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

Show:

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

Show:

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

Show:

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

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