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<h1 class="center">Inverse Sine, Cosine, Tangent</h1>
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<div class="def">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="Right-Angled Triangle" height="120" /></p><h3>Quick Answer: </h3><p>For a <a href="../right_angle_triangle.html">right-angled triangle</a>:</p>
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<div style="clear:both"></div>
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<p class="center"><img src="images/sin-sin-1.svg" alt="sin vs sin-1" /> </p>
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<p> </p>
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<p class="center">The <b>sine</b> function <span class="largest">sin</span> takes angle θ and gives the ratio <span class="intbl"> <em>opposite</em> <strong>hypotenuse </strong></span></p>
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<p class="center">The <b>inverse sine</b> function <span class="largest">sin<sup>-1</sup></span> takes the ratio <span class="intbl"> <em>opposite</em><strong>hypotenuse </strong></span> and gives angle θ</p>
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<p>And cosine and tangent follow a similar idea. </p>
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</div>
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<div class="example">
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<h3>Example (lengths are only to one decimal place):</h3>
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<p style="float:right; margin: 25px 0 25px 10px;"><img src="../geometry/images/triangle-28-40-49.gif" width="159" height="117" alt="triangle 2.8 4.0 4.9 has 35 degree angle" /></p>
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<div class="tbl">
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<div class="row"><span class="left"><b>sin(35°)</b></span><span class="right">= Opposite / Hypotenuse </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 2.8/4.9 </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 0.57...</span></div>
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<div class="row"><span class="left"></span></div>
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<div class="row"><span class="left"><b>sin<sup>-1</sup>(Opposite / Hypotenuse)</b></span><span class="right">= sin<sup>-1</sup>(0.57...) </span></div>
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<div class="row"><span class="left"> </span><span class="right">= 35°</span></div>
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</div>
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</div>
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<h3>And now for the details:</h3>
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<p><a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a> are all based on a Right-Angled Triangle</p>
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<p>They are very similar functions ...
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so we will look at the <b>Sine Function</b> and then <b>Inverse Sine</b> to learn what it is all about.</p>
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<h2>Sine Function</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" /></p>
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<p>The Sine of angle <b><i>θ</i></b> is:</p>
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<ul>
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<li>the <b>length of the side Opposite</b> angle <b><i>θ</i></b></li>
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<li>divided by the <b>length of the Hypotenuse</b></li>
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</ul>
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<p align="left">Or more simply:</p>
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<p align="center" class="larger">sin(<i>θ</i>) = Opposite / Hypotenuse</p>
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<div class="example">
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<h3>Example: What is the sine of 35°?</h3>
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<table width="100%" border="0">
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<tr>
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<td><img src="../geometry/images/triangle-28-40-49.gif" width="159" height="117" alt="triangle 2.8 4.0 4.9 has 35 degree angle" /></td>
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<td><p>Using this triangle (lengths are only to one decimal place):</p>
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<p class="center larger">sin(35°) = Opposite / Hypotenuse <br>
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= 2.8/4.9 <br>
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= <b>0.57...</b></p></td>
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</tr>
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</table>
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</div>
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<p>The Sine Function can help us solve things like this:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-2example2.gif" width="242" height="170" alt="trig ship example 30m at 39 degrees" /> </p>
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<h3>Example: Use the <b>sine function</b> to find <b>"d"</b></h3>
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<p>We know</p>
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<ul>
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<li>The angle the cable makes with the seabed is 39° </li>
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<li>The cable's length is 30 m. </li>
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</ul>
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<p> And we want to know "d" (the distance down).</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">sin 39° = opposite/hypotenuse</span></div>
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<div class="row"><span class="left"> </span><span class="right"> sin 39° = d/30</span></div>
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<div class="row"><span class="left">Swap Sides:</span><span class="right">d/30 = sin 39° </span></div>
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<div class="row"><span class="left">Use a calculator to find sin 39°:</span><span class="right"> d/30 = <span class="hilite">0.6293…</span></span></div>
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<div class="row"><span class="left">Multiply both sides by 30:</span><span class="right">d = 0.6293… x 30 </span></div>
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<div class="row"><span class="left"> </span><span class="right">d = <b>18.88</b> to 2 decimal places</span></div>
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</div>
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<div class="indent50px"></div>
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<p align="center" class="larger"> The depth "d" is <b>18.88 m</b></p>
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</div>
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<h2>Inverse Sine Function</h2>
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<p>But sometimes it is the <b>angle</b> we need to find. </p>
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<p>This is where "Inverse Sine" comes in. </p>
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<p align="center" class="larger">It answers the question "what <b>angle</b> has sine equal to opposite/hypotenuse?"</p>
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<p align="left">The symbol for inverse sine is <b>sin<sup>-1</sup></b>, or sometimes <b>arcsin</b>.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-2example3.gif" width="242" height="170" alt="trig ship example 30m and 18.88m" /> </p>
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<h3>Example: Find the angle <b>"a"</b></h3>
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<p>We know</p>
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<ul>
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<li>The distance down is 18.88 m.</li>
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<li>The cable's length is 30 m. </li>
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</ul>
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<p> And we want to know the angle "a"</p>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">sin a° = opposite/hypotenuse</span></div>
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<div class="row"><span class="left"> </span><span class="right"> sin a° = 18.88/30</span></div>
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<div class="row"><span class="left">Calculate 18.88/30:</span><span class="right">sin a° = 0.6293...</span></div>
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<p>What <b>angle</b> has sine equal to 0.6293...?<br />
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The <b>Inverse Sine</b> will tell us.</p>
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<div class="row"><span class="left">Inverse Sine:</span><span class="right">a° = <b>sin<sup>−1</sup></b>(0.6293...)</span></div>
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<div class="row"><span class="left">Use a calculator to find <b>sin<sup>−1</sup></b>(0.6293...):</span><span class="right">a° = <b>39.0°</b> (to 1 decimal place)</span></div>
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</div>
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<p align="center" class="larger"> The angle "a" is <b>39.0°</b></p>
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</div>
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<h2>They Are Like Forward and Backwards!</h2>
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<div align="center"><img src="images/sin-sin-1.svg" alt="sin vs sin-1" /></div>
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<ul>
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<li><span class="large">sin</span> takes an <b>angle</b> and gives us the <b>ratio</b> "opposite/hypotenuse"</li>
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<li><span class="large">sin<sup>-1</sup></span> takes the <b>ratio</b> "opposite/hypotenuse" and gives us the <b>angle.</b></li>
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</ul>
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<div class="example">
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<h3>Example:</h3>
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<div class="tbl">
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<div class="row"><span class="left">Sine Function:</span><span class="right">sin(<b>30°</b>) = <b>0.5 </b></span></div>
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<div class="row"><span class="left">Inverse Sine:</span><span class="right">sin<sup>−1</sup>(<b>0.5</b>) = <b>30°</b></span></div>
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</div>
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</div>
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<h2 align="left">Calculator</h2>
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<table width="80%" border="0" align="center">
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<tr>
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<td><img src="images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" width="118" height="75" /></td>
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<td>On the calculator you press one of the following (depending on your brand of calculator):
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either '2ndF sin' or 'shift sin'.</td>
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</tr>
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</table>
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<p align="center" class="center80">On your calculator, try using <span class="larger">sin</span> and then <span class="larger">sin<sup>-1</sup></span> to see what happens</p>
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<h2 align="left">More Than One Angle!</h2>
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<p align="left">Inverse Sine <b>only shows you one angle</b> ... but there are more angles that could work.</p>
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<div class="example">
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<h3 align="left">Example: Here are two angles where opposite/hypotenuse = 0.5</h3>
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<p align="center"><br />
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<img src="images/trig-inverse-sin-cos-tan1.gif" width="247" height="79" alt="triangle at 30 and 150 degrees" /></p>
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</div>
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<p align="left">In fact there are <b>infinitely many angles</b>, because you can keep adding (or subtracting) 360°:</p>
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<p align="center"><img src="images/trig-inverse-sin-cos-tan2.svg" alt="sine crosses 0.5 at 30,150,390, etc" style="max-width:100%" /></p>
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<p align="left">Remember this, because there are times when you actually need one of the other angles!</p>
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<h2 align="left">Summary</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="Right-Angled Triangle" height="120" /></p>
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<p>The Sine of angle <b><i>θ</i></b> is:</p>
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<p align="center" class="larger">sin(<i>θ</i>) = Opposite / Hypotenuse</p>
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<p>And Inverse Sine is :</p>
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<p align="center" class="larger">sin<sup>-1</sup> (Opposite / Hypotenuse) = <i>θ </i></p>
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<p> </p>
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<h2>What About "cos" and "tan" ... ?</h2>
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<p>Exactly the same idea, but different side ratios.</p>
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<h4>Cosine</h4>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="Right-Angled Triangle" height="120" /></p>
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<p>The Cosine of angle <b><i>θ</i></b> is:</p>
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<p align="center" class="larger">cos(<i>θ</i>) = Adjacent / Hypotenuse</p>
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<p>And Inverse Cosine is :</p>
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<p align="center" class="larger">cos<sup>-1</sup> (Adjacent / Hypotenuse) = <i>θ </i></p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-example2.gif" width="194" height="124" style="float:left; margin: 10px;" alt="trig example" /></p>
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<h3>Example: Find the size of angle a°</h3>
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<p>cos a° = Adjacent / Hypotenuse</p>
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<p>cos a° = 6,750/8,100 = 0.8333...</p>
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<p>a° = <b>cos<sup>-1</sup></b> (0.8333...) = <b>33.6°</b> (to 1 decimal place)</p>
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</div>
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<h4>Tangent</h4>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="Right-Angled Triangle" height="120" /></p>
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<p>The Tangent of angle <b><i>θ</i></b> is:</p>
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<p align="center" class="larger">tan(<i>θ</i>) = Opposite / Adjacent</p>
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<p>So Inverse Tangent is :</p>
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<p align="center" class="larger">tan<sup>-1</sup> (Opposite / Adjacent) = <i>θ</i></p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-example1.gif" width="220" height="160" alt="trig example" /> </p>
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<h3>Example: Find the size of angle x°</h3>
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<p>tan x° = Opposite / Adjacent </p>
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<p>tan x° = 300/400 = 0.75</p>
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<p>x° = <b>tan<sup>-1</sup></b> (0.75) = <b>36.9°</b> (correct to 1 decimal place)</p>
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</div>
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<p> </p>
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<h2>Other Names</h2>
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<p class="words">Sometimes sin<sup>-1</sup> is called <b>asin</b> or <b>arcsin</b><br />
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Likewise cos<sup>-1</sup> is called <b>acos</b> or <b>arccos</b><br />
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And tan<sup>-1</sup> is called <b>atan</b> or <b>arctan</b></p>
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<div class="example">
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<h3>Examples:</h3>
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<ul>
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<li><b>arcsin(y)</b> is the same as <b>sin<sup>-1</sup>(y) </b><br />
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</li>
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<li><b>atan(θ)</b> is the same as <b>tan<sup>-1</sup>(θ)</b><br />
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</li>
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<li>etc.</li>
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</ul>
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</div>
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<h2>The Graphs</h2>
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<p>And lastly, here are the graphs of Sine, Inverse Sine, Cosine and Inverse Cosine:</p>
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<div class="boxa" style="width: 395px;"><span class="boxa" style="width: 395px;"><img src="images/sine-graph.svg" alt="sine graph" /></span><br />
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<span class="larger">Sine</span></div>
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<div class="boxa" style="width: 195px;"><span class="boxa" style="width: 195px;"><img src="images/inverse-sine-graph.svg" alt="inverse sine graph" /></span><br />
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<span class="larger">Inverse Sine</span></div>
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<div class="boxa" style="width: 395px;"><span class="boxa" style="width: 395px;"><img src="images/cosine-graph.svg" alt="cosine graph" /></span><br />
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<span class="larger">Cosine</span></div>
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<div class="boxa" style="width: 195px;"><span class="boxa" style="width: 195px;"><img src="images/inverse-cosine-graph.svg" alt="inverse cosine graph" /></span><br />
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<span class="larger">Inverse Cosine</span></div>
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<p>Did you notice anything about the graphs? </p>
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<ul>
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<li>They look similar somehow, right?</li>
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<li>But the Inverse Sine and Inverse Cosine don't "go on forever" like Sine and Cosine do ...</li>
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</ul>
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<div class="center80">
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<p>Let us look at the example of Cosine. </p>
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<p>Here is <b>Cosine</b> and <b>Inverse Cosine</b> plotted on the same graph:</p>
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<p align="center"><img src="images/cosine-mirror-graph.svg" alt="cosine mirror graph" /><br>
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<span class="larger">Cosine and Inverse Cosine </span></p>
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<p>They are mirror images (about the diagonal)</p>
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<p>But why does Inverse Cosine get chopped off at top and bottom (the dots are not really part of the function) ... ? </p>
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<p class="center large">Because <a href="../sets/function.html">to be a function</a> it can only give <b>one answer</b> <br>
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when we ask <i>"what is cos<sup>-1</sup>(x) ?"</i> </p>
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<h3>One Answer or Infinitely Many Answers</h3>
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<p>But we saw earlier that there are <b>infinitely many answers</b>, and the dotted line on the graph shows this.</p>
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<p>So yes there <b>are</b> infinitely many answers ...</p>
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<p>... but imagine you type <span class="hilite">0.5</span> into your calculator, press <span class="hilite">cos<sup>-1</sup></span> and it gives you a never ending list of possible answers ... </p>
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<p align="center" class="larger">So we have this rule that <b>a function can only give one answer</b>.</p>
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<p>So, by chopping it off like that we get just one answer, but <b>we should remember that there could be other answers</b>.</p>
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</div>
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<h2>Tangent and Inverse Tangent</h2>
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<p>And here is the tangent function and inverse tangent. Can you see how they are mirror images (about the diagonal) ...? </p>
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<div class="boxa" style="width: 395px;"><span class="boxa" style="width: 395px;"><img src="images/tangent-graph.svg" alt="tangent graph" /></span><span class="larger"><br>
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Tangent</span></div>
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<div class="boxa" style="width: 305px;"><span class="boxa" style="width: 305px;"><img src="images/inverse-tangent-graph.svg" alt="inverse tangent graph" /></span><span class="larger"><br />
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Inverse Tangent</span></div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(3924, 3925, 3926, 3927, 3928, 3929, 3930, 3931, 3932, 3933);</script> </div>
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<div class="related"> <a href="../games/random-trigonometry.html">Random Trigonometry</a> <a href="trig-sine-law.html">The Law of Sines</a> <a href="trig-cosine-law.html">The Law of Cosines</a> <a href="trig-solving-triangles.html">Solving Triangles</a> <a href="trigonometry-index.html">Trigonometry Index</a> <a href="index.html">Algebra Index</a> </div>
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