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      <text>5.3.1.1 Oscillating functions: amplitude, period and frequency</text>
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      <text><![CDATA[<p></p><h4>Oscillating functions: amplitude, period and frequency</h4><p>Particular types of periodic functions that are especially important in engineering are the sine and cosine functions. These are possible choices when modelling behaviour that involves oscillation or motion in a circle. The usefulness of these functions is rather limited if we confine our attention only to \(\sin(x)\) and \(\cos(x)\). Use of functions such as \(3\sin(2x)\), \(5\cos(3x)\) and so on, and other functions made up of sums of functions of this type, enables the modelling of a great variety of situations where the quantity being modelled is known to change in a periodic way. Here we will examine the behaviour of sine and cosine functions and consider a modelling context where choice of a sine function is appropriate. Figure 6 shows how the terms <b>amplitude</b>, <b>period</b>&nbsp;and <b>frequency</b>&nbsp;are defined with respect to a general <b>sinusoid</b>&nbsp;(the name for any general sine or cosine function).&nbsp;<br></p><p><img src="@@PLUGINFILE@@/5-3-fig-1-amp_and_period.svg" alt="" width="461" height="236" role="presentation" class="img-responsive atto_image_button_text-bottom"><br></p><p>Defining amplitude and period for a sinusoid<br></p><p>The amplitude represents the difference between the maximum (or minimum) value of a sinusoidal function and its mean value (which is zero in Figure 6). The frequency represents the number of complete cycles of the function in each unit change in \(x\). The period is such that \(f(x + T) = f(x)\) for all \(x\), e.g. for \(\sin x\), \(T=2\pi\).<br><br>Note that the cosine functions \(\cos nx\) have the same shape as the sine functions \(\sin nx\) but, at \(x = 0\), the cosine functions have a peak or maximum, whereas the sine functions have the value zero, which is the mean value for both of these functions. Indeed the graph of \(y = \cos x\) is exactly like that for \(y= \sin x\) with all the \(x\) values displaced by \(\pi /2\). More general forms of sine and cosine function are given by \(y\) = \(a\sin (bx)\), and \(y\) = \(a\cos(bx)\) where \(a\) and \(b\) are arbitary constants. These are functions with frequency \(\dfrac{b}{2\pi}\), period \(\dfrac{2\pi}{ b}\) and amplitude \(a\). The peak values of the sine functions occur at \(x\) values equal to \(\dfrac{\pi}{2}\), \(\dfrac{5\pi }{2}\), \(\dfrac{9\pi }{2}\) etc. The minimum values occur at \(x\) values equal to \(\dfrac{3\pi }{2}\), \(\dfrac{7\pi }{2}\), \(\dfrac{11\pi }{2}\) etc. When the period is measured in seconds, frequency is measured in cycles per second or Hz which has units of 1/time.<br></p><p></p>]]></text>
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