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It is always possible to write a sum of sinusoidal functions

 (1)

as a single sinusoid the form

 (2)

This can be done by expanding (2) using the trigonometric addition formulas to obtain

 (3)

Now equate the coefficients of (1) and (3)

 (4) (5)

so

 (6) (7)

and

 (8) (9)

giving

 (10) (11)

Therefore,

 (12)

(Nahin 1995, p. 346).

In fact, given two general sinusoidal functions with frequency ,

 (13) (14)

their sum can be expressed as a sinusoidal function with frequency

 (15) (16) (17)

Now, define

 (18) (19)

Then (17) becomes

 (20)

Square and add (◇) and (◇)

 (21)

Also, divide (◇) by (◇)

 (22)

so

 (23)

where and are defined by (◇) and (◇).

This procedure can be generalized to a sum of harmonic waves, giving

 (24) (25)

where

 (26) (27)

and

 (28)

Fourier Series, Prosthaphaeresis Formulas, Simple Harmonic Motion, Sinusoid, Superposition Principle, Trigonometric Addition Formulas, Trigonometry

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## References

Nahin, P. The Science of Radio. Woodbury, NY: American Institute of Physics, 1995.