<h2>Revision of The Tangent Function</h2>
<section>
<h3 class="title">Functions of the form \(y = \tan\theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)</h3>
<img alt="b0314c458ecaf2909d5e63fb0e619ed2.png" src="https://data.ulearngo.com/assets/8debdaa5-1590-4948-a3e0-1c1cab229d6c"/>
<p>The dashed vertical lines are called the asymptotes. The asymptotes are at the values of θ where \(\tan\theta\) is not defined.</p>
<ul>
<li>
<p>Period: \(\text{180}\text{°}\)</p>
</li>
<li>
<p>Domain: \(\{\theta : \text{0}\text{°} \le \theta \le \text{360}\text{°}, \theta \ne \text{90}\text{°}; \text{270}\text{°}\}\)</p>
</li>
<li>
<p>Range: \(\{f(\theta):f(\theta)\in ℝ\}\)</p>
</li>
<li>
<p>\(x\)-intercepts: \((\text{0}\text{°};0)\), \((\text{180}\text{°};0)\), \((\text{360}\text{°};0)\)</p>
</li>
<li>
<p>\(y\)-intercept: \((\text{0}\text{°};0)\)</p>
</li>
<li>
<p>Asymptotes: the lines \(\theta =\text{90}\text{°}\) and \(\theta =\text{270}\text{°}\)</p>
</li>
</ul>
</section>
<section>
<h4 class="title">Functions of the form \(y = a \tan \theta + q\)</h4>
<p>Tangent functions of the general form \(y = a \tan \theta + q\), where \(a\) and \(q\) are constants.</p>
<h3><strong>The effects of \(a\) and \(q\) on \(f(\theta) = a \tan \theta + q\):</strong></h3>
<ul>
<li>
<p><strong>The effect of \(q\) on vertical shift</strong></p>
<ul>
<li>
<p>For \(q&gt;0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.</p>
</li>
<li>
<p>For \(q&lt;0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.</p>
</li>
</ul>
</li>
<li>
<p><strong>The effect of \(a\) on shape</strong></p>
<ul>
<li>
<p>For \(a&gt;1\), branches of \(f(\theta)\) are steeper.</p>
</li>
<li>
<p>For \(0&lt;a&lt;1\), branches of \(f(\theta)\) are less steep and curve more.</p>
</li>
<li>
<p>For \(a&lt;0\), there is a reflection about the \(x\)-axis.</p>
</li>
<li>
<p>For \(-1 &lt; a &lt; 0\), there is a reflection about the \(x\)-axis and the branches of the graph are less steep.</p>
</li>
<li>
<p>For \(a &lt; -1\), there is a reflection about the \(x\)-axis and the branches of the graph are steeper.</p>
</li>
</ul>
</li>
</ul>
<table style="width: 644px;">
<tbody>
<tr>
<td style="width: 126px;"> </td>
<td style="width: 254px;">
<p>\(a&lt;0\)</p>
</td>
<td style="width: 264px;">
<p>\(a&gt;0\)</p>
</td>
</tr>
<tr>
<td style="width: 126px;">
<p>\(q&gt;0\)</p>
</td>
<td style="width: 254px;"><img alt="73b7bdb50e5399566ae55f80007c1111.png" src="https://data.ulearngo.com/assets/904ab46f-b463-4ad5-9846-9ea760b9eae1"/></td>
<td style="width: 264px;"><img alt="9f7d362606019519ade93692d8161ce7.png" src="https://data.ulearngo.com/assets/dd29767d-2416-4aac-8778-5a63ab3ab61d"/></td>
</tr>
<tr>
<td style="width: 126px;">
<p>\(q=0\)</p>
</td>
<td style="width: 254px;"><img alt="4d88551659d54dd8f23f375a43e9cc6b.png" src="https://data.ulearngo.com/assets/7ed5ca2e-88b1-43b3-950a-5a5ae94a2dfb"/></td>
<td style="width: 264px;"><img alt="1c4376363790fc45fefe75c3430c9a04.png" src="https://data.ulearngo.com/assets/c627101f-582b-4d52-b9f7-b4ae39ce37c2"/></td>
</tr>
<tr>
<td style="width: 126px;">
<p>\(q&lt;0\)</p>
</td>
<td style="width: 254px;"><img alt="c31962902b66d146ba0ebecb7dd52d15.png" src="https://data.ulearngo.com/assets/696df576-c80b-4e06-9bb0-e3faae67c564"/></td>
<td style="width: 264px;"><img alt="7aa1b8cd46f11243ddc1d0e318c68c6d.png" src="https://data.ulearngo.com/assets/23ff07a8-c5e7-426e-b74c-5ea8d0f08435"/></td>
</tr>
</tbody>
</table>
</section>
<section></section>

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Revision of The Tangent Function

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Discover how to revise and apply functions like Quadratic, Hyperbolic, Exponential, Sine, Cosine and Tangent, and learn to calculate Average Gradient while sketching graphs and understanding the summary and main ideas.


Revision of The Tangent Function

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