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      <text>23.1.2.1 Periodic functions</text>
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      <text><![CDATA[<h4>Periodic functions</h4>
A function \(f(t)\) is periodic if the function values repeat at regular intervals of the independent variable
\(t\). The regular interval is referred to as the period. An example of such function is given in the Figure below.<p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/23-1-fig-1-periodic-fnc.svg" alt="a general periodic function" width="516" height="139" class="img-responsive atto_image_button_text-bottom"><br></p>
<p></p>
If \(P\) denotes the period we have 
\[ f(t + P) = f(t) \]
for any value of \(t\).
<p></p>
The most obvious examples of periodic functions are the trigonometric functions \( \sin t\) and \( \cos t\), both
of which have period \(2\pi\) (using radian measure as we shall do throughout this Section) (see the Figure below). This follows since
\[ \sin(t + 2\pi) = \sin t \]
and
\[ \cos(t + 2\pi) = \cos t \]<p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/23-1-fig-2-sin-cos.svg" alt="sin and cos" width="540" height="174" class="img-responsive atto_image_button_text-bottom"><br></p>
<p></p>
The <b>amplitude</b> of these sinusoidal functions is the maximum displacement from \(y = 0\) and is clearly \(1\). (Note that we use the term sinusoidal to include cosine as well as sine functions.)
More generally we can consider a sinusoid
\[ y = A \sin nt \]
which has maximum value, or amplitude, \(A\) and where \(n\) is usually a positive integer.
<p></p>
For example
\[ y = \sin 2t \]
is a sinusoid of amplitude \(1\) and period \( \dfrac{2\pi}{2} = \pi \). This is shown in the Figure below.<p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/23-1-fig-3-sin2t.svg" alt="sin2t" width="340" height="214" class="img-responsive atto_image_button_text-bottom"><br></p>
The fact that the period is \(\pi\) follows because
\[ \sin 2(t + \pi) = \sin(2t + 2\pi) = \sin 2t\]
for any value of \(t\).
<p></p>
We see that \(y = \sin 2t\) has half the period of \( \sin t\), \(\pi\) as opposed to \(2\pi\) (see the Figure below).<p><br></p><p><img src="@@PLUGINFILE@@/23-1-fig-4-sint-sin2t-comp.svg" alt="sint-sin2t-comp" width="340" height="200" class="img-responsive atto_image_button_text-bottom"><br></p><p>&nbsp;
This can
alternatively be phrased by stating that \(\sin 2t\) oscillates twice as rapidly (or has twice the frequency)
of \(\sin t\).</p>
<p></p>
In general, \(y = A\sin nt\) has amplitude \(A\), period \(\dfrac{2\pi}{n}\) and completes \(n\) oscillations when \(t\) changes by \(2\pi\). Formally, we define the <b>frequency</b> of a sinusoid as the reciprocal of the period:
<p></p>
\[ \text{frequency} = \dfrac{1}{\text{period}} \]
and the <b>angular frequency</b>, often denoted the Greek Letter \(\omega\) (omega) as
\[ \text{angular frequency} = 2 \pi \times \text{frequency} = \dfrac{2\pi}{\text{period}} .\]
Thus \(y = A\sin nt\) has frequency \( \dfrac{n}{2\pi}\) and angular frequency \(n\).]]></text>
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