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      <text>5.3.6 (Plotting trig graphs (sine))</text>
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      <text><![CDATA[<h1>Plotting trig functions</h1>

<p>Just as we have seen when plotting quadratics, exponentials, logarithms and polynomials, it is often useful to be able to pass between the algebraic form of a function and the graph of a function. In this section we will see how to plot the sine, cosine and tangent functions and we will learn the general forms of such functions and how we can transform them.</p>
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<h3>The general sine function</h3>
<p>Since we have defined \(\sin x\) in terms of the coordinates of a point on a circle of unit radius it follows immediately that \(-1\le\sin x \le +1\qquad\text{and}\qquad -1\le \cos x \le +1\) for any value of \(x\). Therefore the amplitude of the sine and cosine graphs is \(1\) and the principal axis is the line \(y = 0\).</p>
<p> We can draw the graph of the sine function, see the figure below. We have labelled the horizontal axis using degrees and have shown two periods. Notice the graph of \(\sin(x)\) repeats every \(360^\circ\), with maximum values at \(90^\circ+360k^\circ\) and minimum values at \(270^\circ+360k^\circ\) for \(k \in \mathbb{N}\). In addition, notice that \(f(x) = \sin x\) is an <i>odd</i> function because \(f(-x) = -f(x)\).</p>
<p><strong><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Labelled%20Sine%20wave.png" alt="labelled diagram of a sine function" width="398" height="229" /></strong></p>

<p><strong>The general sine function</strong> is the function: \[ y = a \sin b(x-c) + d. \] We will see that \(a\) affects the amplitude of the graph, \(b\) affects the period, \(c\) affects the horizontal translation and \(d\) affects vertical translation. Move the sliders below to see what effect altering each of these values has on the sine function.</p>

<iframe scrolling="no" title="Sine sliders" src="https://www.geogebra.org/material/iframe/id/xyrw4kdj/width/1920/height/964/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" width="900px" height="450px" style="border:0px;"> </iframe>]]></text>
<file name="4.2 fig 11 graphs of trigs.svg" path="/" encoding="base64">[encoded data, 32180 characters]</file>
<file name="Labelled Sine wave.png" path="/" encoding="base64">[encoded data, 188380 characters]</file>
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