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# Sums of sinusoids¶

This page largely based on http://math.stackexchange.com/a/1239123 with thanks.

## Sinusoids¶

Paraphrasing Wolfwram mathworld - a *sinusoid* is a function of
some variable, say \(t\), that is similar to the sine function but may be
shifted in phase, frequency, amplitude, or any combination of the three.

The general formula for a sinusoid function is:

where:

- \(A\) is the
*amplitude*— the maximum value of the function; - \(f\) is the
*ordinary frequency*— the number of cycles per unit of \(t\); - \(\omega = 2 \pi f\) is the
*angular frequency*— the number of radians per unit of \(t\); - \(\theta\) is the
*phase offset*(in radians).

The standard sine function \(f(t) = \sin(t)\) is a special case of a sinusoid, with \(A = 1\), \(f = 1 / 2 \pi\), \(\theta = 0\).

The standard cosine function \(f(t) = \cos(t)\) is a special case of a sinusoid, with \(A = 1\), \(f = 1 / 2 \pi\), \(\theta = -pi / 2\).

## The sum of sinusoids with the same frequency is also a sinusoid¶

Remembering The angle sum rule, we can write any sinusoid as a weighted sum of a sine and a cosine:

where:

Equation (2) also points us to the fact that any weighted sum of a sine and cosine can be written as a single sinusoid. For any \(A', A''\), we can solve equations (3) for \(A, \theta\):

Thus, any sum of sinusoids, *of the same frequency* and therefore the same
input \(\omega t\), is also a sinusoid: