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      <text>4.1.2.1 Solving right-angled triangles</text>
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      <text><![CDATA[<h3> Solving right-angled triangles<br></h3>

<p>Solving right-angled triangles means obtaining the values of all the angles and all the sides of a given right-angled triangle using the trigonometric functions (and, if necessary, the inverse trigonometric functions) and perhaps Pythagoras' theorem.</p>
<p>There are three cases to be considered:</p>
<p><br></p>

<p><strong>Case 1: Given the hypotenuse and an angle</strong></p>
<p>We use \(\sin\) or \(\cos\) as appropriate:</p>
<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/Trig%286%29Fig6%28ab%29.svg" alt="A right angled triangle ABC, with  hypotenuse h, x is adjacent side to angle A and y is opposite side to angle A" width="200" height="174" class="img-responsive atto_image_button_text-bottom">
</figure>

<p>Assuming \(h\) and \(\theta\) in the previous figure are given then</p>
<p>\(\qquad \cos\theta = \dfrac{x}{h}\) which gives \(x = h \cos\theta\)</p>
<p>from which \(x\) can be calculated.</p>
<p>Also</p>
<p>\(\qquad \sin\theta = \dfrac{y}{h}\) so \(y = h \sin\theta\) which enables us to calculate \(y.\)</p>
<p>Clearly the third angle of this triangle (at \(B\)) is \(90^\circ-\theta.\)</p>
<p><br></p>

<p><strong>Case 2: Given a side other than the hypotenuse and an angle.</strong></p>
<p>We use \(\tan\):</p>
<p>(a) If \(x\) and \(\theta\) are known then, in previous figure, \(\quad \tan\theta = \dfrac{y}{x}\quad \text{so}\quad y = x\tan\theta\)</p>
<p>which enables us to calculate \(y.\)</p>
<p>(b) If \(y\) and \(\theta\) are known then \( quad \qquad \tan\theta = \dfrac{y}{x} \qquad \text{gives} \qquad x = \dfrac{y}{\tan\theta}\quad\) from which \(x\) can be calculated.</p>
<p>Then the hypotenuse can be calculated using Pythagoras' theorem: \( \qquad h=\sqrt{x^2+y^2}\)</p>
<p><br></p>

<p><strong>Case 3: Given two of the sides</strong></p>
<p>We use \(\tan^{-1}\) or \(\sin^{-1}\) or \(\cos^{-1}\):</p>
<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/Trig%286%29Fig6%28c%29.svg" alt="A right angle triangle where we are given the adjacent and oposite side. " width="500" height="121" class="img-responsive atto_image_button_text-bottom">
</figure>

<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/Trig%286%29Fig6%28d%29.svg" alt="A right angle triangle where we are given the opposite side and the  hypotenuse" width="464" height="105" class="img-responsive atto_image_button_text-bottom">
</figure>

<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/Trig%286%29Fig6%28e%29.svg?time=1649852796159" alt="A right angle triangle where we are given the adjacent side and the hypotenuse." width="500" height="129" class="img-responsive atto_image_button_text-bottom">
</figure>

<p>Note: since two sides are given we can use Pythagoras' theorem to obtain the length of the third side at the outset.</p>]]></text>
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