Harmonic Addition Theorem -- from Wolfram MathWorld

The harmonic addition theorem states that, for real numbers , , and , the sum

(1)

can be expressed as a single sinusoid of the form

(2)

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

(3)

Now equate the coefficients in the two expressions for

			(4)
			(5)

so, when ,

			(6)
			(7)

and

			(8)
			(9)

For , a consistent choice is

			(10)
			(11)

Here is the principal inverse tangent, and is the sign function. Therefore, for ,

(12)

(Nahin 1995, p. 346). If and , take and . If , take and any real .

In fact, given two general sinusoidal functions with real coefficients and phases, and a common real frequency ,

			(13)
			(14)

their sum can be expressed as a sinusoid with frequency

			(15)
			(16)
			(17)

Now, define

			(18)
			(19)

Then the sum becomes

(20)

Square and add (18) and (19)

(21)

When , divide (19) by (18)

(22)

(23)

where is the nonnegative square root of the right-hand side of (21), and satisfies both (18) and (19). These two relations determine the phase modulo when , including cases where the tangent formula is undefined. When , the sum is identically zero and the phase is arbitrary.