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

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Sinusoids
*********

Paraphrasing `Wolfwram mathworld
<http://mathworld.wolfram.com/Sinusoid.html>`_ - 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
<https://en.wikipedia.org/wiki/Sine_wave>`_ is:

.. math::
    :label: sin_sinusoid

    f(t) = A \sin(2 \pi f t + \theta) = A \sin(\omega t + \theta)

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 :doc:`angle_sum`, we can write any sinusoid as a weighted sum of a
sine and a cosine:

.. math::
    :label: sinusoid_as_sum

    A \sin(\omega t + \theta) = \\
    A \sin(\omega t) \cos(\theta) + A \cos(\omega t) \sin(\theta) = \\
    A' \sin(\omega t) + A'' \cos(\omega t)

where:

.. math::
    :label: a_dash_dash

    A' = A \cos(\theta) \\
    A'' = A \sin(\theta)

Equation :eq:`sinusoid_as_sum` 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 :eq:`a_dash_dash` for $A, \theta$:

.. math::

    A = \sqrt{A'^2 + A''^2} \\
    \tan(\theta) = \frac{A''}{A'}

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

.. math::

    A \sin(\omega t + \theta) + B \sin(\omega t + \phi) =
    (A' + B') \sin(\omega t) + (A'' + B'') \cos(\omega t)

.. include:: links_names.inc