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      <text>5.3.8 (Transformations of sine and cosine)</text>
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    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Transformations of sines and cosines</h3>

<p>We have seen that the forms of the general sine and cosine functions are similar. This means that for both \(y = a \sin b(x-c) + d\) and \(y = a \cos b(x-c) + d\), the coefficients \(a,b,c,d\) control the same properties. Below we outline the specific effect each of the coefficients has on these functions.</p>

<ol>
<li>The <i>amplitude</i> is \(|a|\).</li>
<li>The <i>period</i> is \(\frac{360^\circ}{b}\), for \(b > 0\).</li>
<li> The <i>principal axis</i> is \(y = d\).</li>
<li>\(y = a \sin b(x-c) + d\) is obtained from \(y = \sin x\) by a vertical dilation of factor \(a\) and a horizontal dilation with factor \(\frac{1}{b}\), followed by a horizontal translation of \(c\) units and a vertical translation of \(d\) units.</li>
<li>\(y = a \cos b(x-c) + d\) is obtained from \(y = \cos x\) by a vertical dilation of factor \(a\) and a horizontal dilation with factor \(\frac{1}{b}\), followed by a horizontal translation of \(c\) units and a vertical translation of \(d\) units.</li>
</ol>]]></text>
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