<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>5.0.1 (The unit circle)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h1>The unit circle</h1>

<p>You may be most familiar with the definitions of trigonometric functions via use of right angled triangles. However, it is possible to define the trigonometric functions in greater generality. We do this by using the <i>unit circle</i>. The <b>unit circle</b> is the circle with centre at the origin and a radius of \(1\) unit. The equation of the unit circle is \(x^2+y^2 = 1\).</p>

<p>Suppose that in addition, we have an arrow denoted \(OP\) from the origin to a point \(P\) on the circumference of this circle. We are then interested in the angle \(\theta\), that is the angle the arrow makes with the positive \(x\)-axis. See the figure below<br/></p>

<p style="text-align: center;"><img src="@@PLUGINFILE@@/4.1%20fig%202%20angle%20theta.svg" alt="Clockwise angle from origin to point P on unit circle" role="presentation" class="img-responsive atto_image_button_text-bottom" width="280" height="227"><br></p>

<p>Imagine that \(OP\) rotates in anti-clockwise direction. With this sense of rotation the angle \(\theta\) is taken as positive whereas a <i>clockwise rotation</i> is taken as negative. See examples below.</p>

<figure><p style="text-align: center;"><img src="@@PLUGINFILE@@/4.2%20fig3clockwiserotation1.1.svg" alt="Angle theta of 90 degrees, so the vector points up" class="img-responsive atto_image_button_text-bottom" width="178" height="178"></p>
<p style="text-align: center;"> 
\(\theta = 90^\circ\), a quarter rotation anticlockwise</p><p style="text-align: center;">
<img src="@@PLUGINFILE@@/4.2%20fig3clockwiserotation2.svg" alt="Angle theta of 315 degrees, so the vector P points southeast" class="img-responsive atto_image_button_text-bottom" width="178" height="178"></p>
<p style="text-align: center;">\(\theta = 315^\circ\), a seven eighths rotation anticlockwise</p>
<p style="text-align: center;"><img src="@@PLUGINFILE@@/4.2%20fig3clockwiserotation3.2.svg" alt="Angle theta of -45 degrees, so the vector P points southeast" class="img-responsive atto_image_button_text-bottom" width="178" height="178"></p>
<p style="text-align: center;">\(\theta = -45^\circ\), a one eighths rotation clockwise</p></figure>
<p style="text-align: center;"><br></p>]]></text>
<file name="4.1 fig 2 angle theta.svg" path="/" encoding="base64">[encoded data, 4408 characters]</file>
<file name="4.2 fig 4 first quadrant.svg" path="/" encoding="base64">[encoded data, 8044 characters]</file>
<file name="4.2 fig3clockwiserotation1.1.svg" path="/" encoding="base64">[encoded data, 10644 characters]</file>
<file name="4.2 fig3clockwiserotation2.svg" path="/" encoding="base64">[encoded data, 10352 characters]</file>
<file name="4.2 fig3clockwiserotation3.2.svg" path="/" encoding="base64">[encoded data, 10004 characters]</file>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
