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      <text>4.2.2.1 General definitions of trigonometric functions</text>
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      <text><![CDATA[<h3> General definitions of trigonometric functions </h3>

<p>We now define the trigonometric functions in a more general way than in terms of ratios of sides of a right-angled triangle. To do this we consider a circle of <span><strong>unit radius</strong></span> whose centre is at the origin of a Cartesian coordinate system and an arrow (or <span><strong>radius vector</strong></span>) <span class="math inline">\(OP\)</span> from the centre to a point <span class="math inline">\(P\)</span> on the circumference of this circle. We are interested in the angle <span class="math inline">\(\theta\)</span> that the arrow makes with the <span><strong>positive</strong></span> <span class="math inline">\(x\)</span>-axis. See the figure below</p>
<figure class="HELM_figure">
<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">
</figure>
<p>Imagine that the vector <span class="math inline">\(OP\)</span> rotates in <span><strong>anti-clockwise direction</strong></span>. With this sense of rotation the angle <span class="math inline">\(\theta\)</span> is taken as positive whereas a <span><strong>clockwise</strong></span> rotation is taken as negative. See examples below.</p>
<figure class="HELM_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;"> \(\theta = 90^\circ = \dfrac{\pi}{2} \text{ rad}\), 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;">\(\theta = 315^\circ = \dfrac{7\pi}{4}\text{ rad}\), 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;">\(\theta = -45^\circ = -\dfrac{\pi}{4} \text{ rad}\), a one eighths rotation clockwise</p></figure>

<h4> The sine and cosine of an angle </h4>

<p>For <span class="math inline">\(0\le\theta\le \dfrac{\pi}{2}\)</span> (called the <span><strong>first</strong></span> quadrant) we have the following situation with our unit radius circle. In the figure below we let \(O\) be the origin point, \(P\) a point on the circumference, and \(Q\) and \(R\) the horizontal and vertical projections onto the axes of \(P\), respectively.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/4.2%20fig%204%20first%20quadrant.svg" alt="Vector rotated approximately 45 degrees or pi over 4 radians" role="presentation" class="img-responsive atto_image_button_text-bottom" width="143" height="123">
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<p>The <span><strong>projection</strong></span> of <span class="math inline">\(OP\)</span> along the positive <span class="math inline">\(x\)</span>-axis is <span class="math inline">\(OQ\)</span>. But, in the right-angled triangle <span class="math inline">\(OPQ\)</span>  we have:
</p>
<p>
<span class="math inline"> \(\cos\theta=\frac{OQ}{OP}\qquad \text{or} \qquad OQ=OP\cos\theta\)</span></p>

<p>and since <span class="math inline">\(OP\)</span> has unit length <span class="math inline"> \(\cos\theta =OQ \qquad (1) \)</span></p>

<p>Similarly in this right-angled triangle: </p>

<p> <span class="math inline">\(\sin\theta=\frac{PQ}{OP}\qquad\text{or}\qquad PQ=OP\sin\theta\)</span></p>

<p>but <span class="math inline">\(PQ=OR\)</span> and <span class="math inline">\(OP\)</span> has unit length, so \(\sin\theta=OR\qquad \)
</p>
<p> Equation (1) tells us that we can interpret <span class="math inline">\(\cos\theta\)</span> as the projection of <span class="math inline">\(OP\)</span> along the <span><strong>positive</strong></span> <span class="math inline">\(x\)</span><span><strong>-axis</strong></span> and <span class="math inline">\(\sin \theta\)</span> as the projection of <span class="math inline">\(OP\)</span> along the <span><strong>positive</strong></span> <span class="math inline">\(y\)</span><span><strong>-axis</strong></span>.</p>

<p>We shall use these interpretations as the <span><strong>definitions</strong></span> of <span class="math inline">\(\sin\theta\)</span> and <span class="math inline">\(\cos\theta\)</span> for <span><strong>any</strong></span> values of <span class="math inline">\(\theta\)</span>.</p>

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>For a radius vector <span class="math inline">\(OP\)</span> of a circle of unit radius making an angle <span class="math inline">\(\theta\)</span> with the positive <span class="math inline">\(x\)</span>-axis
<br> <span class="math inline">\(\cos\theta=\)</span> projection of <span class="math inline">\(OP\)</span> along the positive <span class="math inline">\(x\)</span>-axis
<br><span class="math inline">\(\sin\theta=\)</span> projection of <span class="math inline">\(OP\)</span> along the positive <span class="math inline">\(y\)</span>-axis</p>
</div>]]></text>
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