Harmonic Addition Theorem

To convert an equation of the form

$$f(\theta)=a\cos\theta+b\sin\theta$$ (1)

to the form

$$f(\theta)=c\cos(\theta+\delta),$$ (2)

expand (2) using the trigonometric addition formulas to obtain

$$f(\theta) = c\cos\theta\cos\delta-c\sin\theta\sin\delta.$$ (3)

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

$$a = c\cos\delta$$ (4)

$$b = -c\sin\delta,$$ (5)

so

$$\tan\delta = - {b\over a}$$ (6)

$$a^2+b^2 = c^2,$$ (7)

and we have

$$\delta = \tan^{-1}\left({- {b\over a}}\right)$$ (8)

$$c = \sqrt{a^2+b^2}.$$ (9)

Given two general sinusoidal functions with frequency $\omega$:

$$\psi_1 = A_1\sin(\omega t+\delta_1)$$ (10)

$$\psi_2 = A_2\sin(\omega t+\delta_2),$$ (11)

their sum $\psi$ can be expressed as a sinusoidal function with frequency $\omega$

$$\psi \equiv \psi_1+\psi_2= A_1[\sin(\omega t)\cos\delta_1+\sin\delta_1\cos(\omega t)]+A_2[\sin(\omega t)\cos\delta_2+\sin\delta_2\cos(\omega t)]$$

$$= [A_1\cos\delta_1+A_2\cos\delta_2]\sin(\omega t)+[A_1\sin\delta_1+A_2\sin\delta_2]\cos(\omega t).$$ (12)

Now, define

$$A\cos\delta \equiv A_1\cos\delta_1+A_2\cos\delta_2$$ (13)

$$A\sin\delta \equiv A_1\sin\delta_1+A_2\sin\delta_2.$$ (14)

Then (12) becomes

$$A\cos\delta \sin(\omega t)+A\sin\delta\cos(\omega t) = A\sin(\omega t+\delta).$$ (15)

Square and add (13) and (14)

$$A_2 = {A_1}^2+{A_2}^2+2A_1A_2\cos(\delta_2-\delta_1).$$ (16)

Also, divide (14) by (13)

$$\tan\delta = {A_1\sin\delta_1+A_2\sin\delta_2 \over A_1\cos\delta_1+A_2\cos\delta_2},$$ (17)

so

$$\psi = A\sin(\omega t+\delta),$$ (18)

where $A$ and $\delta$ are defined by (16) and (17).

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

$$\psi = \sum_{i=1}^n A_i\cos (\omega t+\delta_i)= A\cos (\omega t+\delta),$$ (19)

where

$$A^2 \equiv \sum_{i=1}^n\sum_{j=1}^n A_iA_j\cos(\delta_i-\delta_j)$$ (20)

$$= \sum_{i=1}^n {A_i}^2 + 2\sum_{i=1}^n \sum_{j>i}^n A_iA_j\cos(\delta_i-\delta_j)$$ (21)

and

$$\tan\delta = {\sum_{i=1}^n A_i\sin \delta_i\over \sum_{i=1}^n A_i\cos \delta_i}.$$ (22)