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<h1 align="center"> Trigonometric Identities<br />
</h1>
<h3 align="center">You might like to read about <a href="trigonometry.html">Trigonometry</a> first!</h3>
<h2>Right Triangle</h2>
<p>The <b>Trigonometric Identities</b> are equations that are true for <a href="../right_angle_triangle.html">Right Angled Triangles</a>. <i>(If it is not a Right Angled Triangle go to the <a href="triangle-identities.html">Triangle Identities</a> page.)</i></p>
<p>Each side of a <b>right triangle</b> has a name:<br />
</p>
<p class="center"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /><br />
</p>
<div class="example">
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse-rot.svg" alt="examples of Opposite, Adjacent and Hypotenuse" /></p>
<p><b>Adjacent</b> is always next to the angle</p>
<p>And <b>Opposite</b> is opposite the angle</p>
</div>
<div class="def">
<p>We are soon going to be playing with all sorts of functions, but remember it all comes back to that simple triangle with:</p>
<ul>
<li>Angle <b>&theta;</b></li>
<li>Hypotenuse</li>
<li>Adjacent</li>
<li>Opposite</li>
</ul>
</div>
<h2>Sine, Cosine and Tangent</h2>
<p>The three main functions in trigonometry are <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a>. </p>
<p align="center">They are just the <b>length of one side
divided by another</b></p>
<p> For a right triangle with an angle <b><i>&theta;</i></b> :</p>
<p class="center"><img src="images/sin-cos-tan.svg" alt="sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent" style="max-width:100%" /></p>
<table border="0" align="center" cellpadding="5">
<tr>
<td><div align="right">Sine Function:</div></td>
<td nowrap><b>sin(<i>&theta;</i>) = Opposite / Hypotenuse</b></td>
</tr>
<tr>
<td><div align="right">Cosine Function:</div></td>
<td nowrap><b>cos(<i>&theta;</i>) = Adjacent / Hypotenuse</b></td>
</tr>
<tr>
<td><div align="right">Tangent Function:</div></td>
<td nowrap><b>tan(<i>&theta;</i>) = Opposite / Adjacent</b></td>
</tr>
</table>
<p class="center">For a given angle <b><i>&theta;</i></b> each ratio stays the same <br>
no matter how big or small the triangle is</p>
<p>&nbsp;</p>
<p>When we divide Sine by Cosine we get:</p>
<p class="center larger"><span class="intbl"><em>sin(&theta;)</em><strong>cos(&theta;)</strong></span> = <span class="intbl"><em>Opposite/Hypotenuse</em><strong>Adjacent/Hypotenuse</strong></span> = <span class="intbl"><em>Opposite</em><strong>Adjacent</strong></span> = tan(&theta;) </p>
<p>So we can say:</p>
<div class="def"><p class="center large">tan(&theta;) = <span class="intbl"><em>sin(&theta;)</em><strong>cos(&theta;)</strong></span></p>
</div>
<p>That is our first <b>Trigonometric Identity</b>.</p>
<h2>Cosecant, Secant and Cotangent</h2>
<p>We can also divide &quot;the other way around&quot; (such as <b>Adjacent/Opposite</b> instead of <b>Opposite/Adjacent</b>):</p>
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /></p>
<table border="0" align="center" cellpadding="5">
<tr>
<td><div align="right">Cosecant Function:</div></td>
<td nowrap><b>csc(<i>&theta;</i>) = Hypotenuse / Opposite</b></td>
</tr>
<tr>
<td><div align="right">Secant Function:</div></td>
<td nowrap><b>sec(<i>&theta;</i>) = Hypotenuse / Adjacent</b></td>
</tr>
<tr>
<td><div align="right">Cotangent Function:</div></td>
<td nowrap><b>cot(<i>&theta;</i>) = Adjacent / Opposite</b></td>
</tr>
</table>
<p>&nbsp;</p>
<div class="example">
<h3>Example: when Opposite = 2 and Hypotenuse = 4 then</h3>
<p align="center"> <b>sin(&theta;) = 2/4</b>, and <b>csc(&theta;) = 4/2</b></p>
</div>
<p>Because of all that we can say:</p>
<div class="def">
<p class="center large">sin(&theta;) = 1/csc(&theta;)</p>
<p class="center large">cos(&theta;) = 1/sec(&theta;)</p>
<p class="center large"> tan(&theta;) = 1/cot(&theta;)<b></b><br>
</p>
</div>
<p>And the other way around:</p>
<div class="def">
<p class="center large">csc(&theta;) = 1/sin(&theta;)</p>
<p class="center large">sec(&theta;) = 1/cos(&theta;)</p>
<p class="center large"> cot(&theta;) = 1/tan(&theta;)<br>
</p>
</div>
<p>And we also have:</p>
<div class="def">
<p class="center large">cot(&theta;) = cos(&theta;)/sin(&theta;)<br>
</p>
</div>
<h2>Pythagoras Theorem</h2>
<p>For the next trigonometric identities we start with <a href="../pythagoras.html">Pythagoras' Theorem</a>: </p>
<table border="0">
<tr>
<td><img src="../geometry/images/triangle-abc.svg" alt="right angled triangle abc" /></td>
<td><p>The Pythagorean Theorem says that, <i>in a right triangle,</i> the square of a plus the square of b is equal to the square of c:</p>
<p align="center" class="large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></p></td>
</tr>
</table>
<p>Dividing through by <i>c</i><sup>2</sup> gives</p>
<p class="center large"><span class="intbl">
<em>a<sup>2</sup></em>
<strong>c<sup>2</sup></strong>
</span> + <span class="intbl">
<em>b<sup>2</sup></em>
<strong>c<sup>2</sup></strong>
</span> = <span class="intbl">
<em>c<sup>2</sup></em>
<strong>c<sup>2</sup></strong>
</span></p>
<p>This can be simplified to:</p>
<p class="center"><span class="large" style="font-size: 150%">(</span><span class="intbl" style="transform: translateY(-10%);">
<em>a</em>
<strong>c</strong>
</span><span class="large" style="font-size: 150%">)<sup>2</sup></span> + <span class="large" style="font-size: 150%">(</span><span class="intbl" style="transform: translateY(-10%);">
<em>b</em>
<strong>c</strong>
</span><span class="large" style="font-size: 150%">)<sup>2</sup> = 1</span> </p>
<p>Now, <b>a/c</b> is <b>Opposite / Hypotenuse</b>, which is <b>sin(&theta;)</b></p>
<p>And <b>b/c</b> is <b>Adjacent / Hypotenuse</b>, which is <b>cos(&theta;)</b></p>
<p>So (a/c)<sup>2</sup> + (b/c)<sup>2</sup> = 1 can also be written:</p>
<div class="def">
<p class="center large">sin<sup>2</sup> &theta; + cos<sup>2</sup> &theta; = 1 </p>
</div>
<div class="def"> Note:
<ul>
<li><b>sin<sup>2</sup> &theta;</b> means to find the sine of &theta;, <b>then</b> square the result, and</li>
<li><b>sin &theta;<sup>2</sup></b> means to square &theta;, <b>then</b> do the sine function</li>
</ul>
</div>
<br />
<div class="example">
<h3>Example: 32&deg;</h3>
<p>Using <b>4 decimal places only</b>:</p>
<ul>
<li>sin(32&deg;) = 0.5299...</li>
<li>cos(32&deg;) = 0.8480...</li>
</ul>
<p>Now let's calculate <b>sin<sup>2 </sup>&theta; + cos<sup>2</sup> &theta;</b>:</p>
<p align="center">0.5299<sup>2</sup> + 0.8480<sup>2</sup> <br>
= 0.2808... + 0.7191... <br>
= <b>0.9999...</b></p>
<p>We get very close to 1 using only 4 decimal places. Try it on <i>your</i> calculator, you might get better results!</p>
</div>
<p>Related identities include:</p>
<div class="def">
<p class="larger center">sin<sup>2</sup> &theta; = 1 − cos<sup>2</sup> &theta;<br>
cos<sup>2</sup> &theta; = 1 − sin<sup>2</sup> &theta;<br>
tan<sup>2</sup> &theta; + 1 = sec<sup>2</sup> &theta;<br>
tan<sup>2</sup> &theta; = sec<sup>2</sup> &theta; − 1<br>
cot<sup>2</sup> &theta; + 1 = csc<sup>2</sup> &theta;<br>
cot<sup>2</sup> &theta; = csc<sup>2</sup> &theta; − 1</p>
</div>
<table border="0" align="center">
<tr>
<td><h2>How Do You Remember Them? </h2>
<p>The identities mentioned so far can be remembered <br />
using one clever diagram called the <a href="trig-magic-hexagon.html">Magic Hexagon</a>:</p>
<p>&nbsp;</p></td>
<td>&nbsp;</td>
<td><a href="trig-magic-hexagon.html"><img src="images/magic-hexagon-1.gif" width="173" height="132" alt="magic hexagon" /></a></td>
</tr>
</table>
<h2>But Wait ... There is More!</h2>
<p>There are many more identities ... here are some of the more useful ones:</p>
<h3>Opposite Angle Identities</h3>
<div class="def">
<p class="larger center">sin(&minus;&theta;) = &minus;sin(&theta;)</p>
<p class="larger center">cos(&minus;&theta;) = cos(&theta;)</p>
<p class="larger center">tan(&minus;&theta;) = &minus;tan(&theta;)</p>
</div>
<h3> Double Angle Identities </h3>
<table border="0" align="center">
<tr>
<td><img src="images/trig-sin2theta.gif" alt="sin 2a" width="165" height="71" /></td>
</tr>
<tr>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/trig-cos2a.gif" alt="cos 2a" width="190" height="134" /></td>
</tr>
<tr>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/trig-tan2a.gif" alt="tan 2a" width="163" height="42" /></td>
</tr>
</table>
<p>&nbsp;</p>
<h3> Half Angle Identities </h3>
<p>Note that &quot;<span class="large">&plusmn;</span>&quot; means it may be <b>either one</b>, depending on the value of <i>&theta;/2</i></p>
<table border="0" align="center">
<tr>
<td><img src="images/trig-sin-half.gif" alt="sin a/2" width="180" height="51" /></td>
</tr>
<tr>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/trig-cos-half.gif" alt="cos a/2" width="182" height="51" /></td>
</tr>
<tr>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/trig-tan-half.gif" alt="tan a/2" style="max-width:100%" /></td>
</tr>
<tr>
<td>&nbsp;</td>
</tr>
<tr>
<td><img src="images/trig-cot-half.gif" alt="cot a/2" style="max-width:100%" /></td>
</tr>
</table>
<br />
<h3>Angle Sum and Difference Identities </h3>
<p>Note that <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> means you can use plus or minus, and the <img src="../images/symbols/minus-plus.svg" alt="minus/plus"> means to use the opposite sign.</p>
<p class="center">sin(A <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> B) = sin(A)cos(B) <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> cos(A)sin(B)</p>
<p class="center">cos(A <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> B) = cos(A)cos(B) <img src="../images/symbols/minus-plus.svg" alt="minus/plus"> sin(A)sin(B)</p>
<p class="center">tan(A <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> B) = <span class="intbl"><em>tan(A) <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> tan(B)</em><strong>1 <img src="../images/symbols/minus-plus.svg" alt="minus/plus"> tan(A)tan(B)</strong></p>
<p class="center">cot(A <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> B) = <span class="intbl"><em>cot(A)cot(B) <img src="../images/symbols/minus-plus.svg" alt="minus/plus"> 1</em><strong>cot(B) <img src="../images/symbols/plus-minus.svg" alt="plus/minus"> cot(A)</strong></p>
<h2> Triangle Identities </h2>
<p>There are also <a href="triangle-identities.html">Triangle Identities</a> which apply to all triangles (not just Right Angled Triangles)</p>
<p>&nbsp;</p>
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