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379 lines
15 KiB
HTML
379 lines
15 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/sine-cosine-tangent.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:00:04 GMT -->
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Sine, Cosine and Tangent</h1>
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<p class="center"><i>Three Functions, but same idea.</i></p>
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<h2>Right Triangle</h2>
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<p>Sine, Cosine and Tangent are the main functions used in <a href="algebra/trigonometry-index.html">Trigonometry</a> and are based on a <a href="right_angle_triangle.html">Right-Angled Triangle</a>.</p>
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<p>Before getting stuck into the functions, it helps to give a <b>name</b> to each side of a right triangle:</p>
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<p class="center"><img src="algebra/images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="190" width="326"></p>
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<div class="bigul">
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<ul>
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<li>"Opposite" is opposite to the angle θ</li>
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<li>"Adjacent" is adjacent (next to) to the angle θ</li>
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<li>"Hypotenuse" is the long one</li>
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</ul>
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</div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="algebra/images/adjacent-opposite-hypotenuse-rot.svg" alt="examples of Opposite, Adjacent and Hypotenuse" height="98" width="378"></p>
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<p><b>Adjacent</b> is always next to the angle</p>
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<p>And <b>Opposite</b> is opposite the angle</p>
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</div>
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<h2>Sine, Cosine and Tangent</h2>
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<p><b>Sine</b>, <b>Cosine</b> and <b>Tangent</b> (often shortened to <b>sin</b>, <b>cos</b> and <b>tan</b>) are each a <b>ratio of sides</b> of a right angled triangle:</p>
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<p class="center"><img src="algebra/images/sin-cos-tan.svg" alt="sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent" style="max-width:100%" height="181" width="468"></p>
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<p class="center">For a given angle <b><i>θ</i></b> each ratio stays the same<br>
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no matter how big or small the triangle is</p>
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<p>To calculate them:</p>
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<p class="center"><span class="large"><b>Divide the length of one side by another side</b>
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</span></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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<p style="float:left; margin: 0 20px 5px 0;"><img src="geometry/images/triangle-28-40-49.gif" alt="triangle with 2.8, 4.0 and 4.9 sides" height="117" width="159"></p>
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<p>Using this triangle (lengths are only to one decimal place):</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><span class="large">sin(35°) </span></td>
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<td><span class="large">= <span class="intbl"><em>Opposite</em><strong>Hypotenuse</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl"><em>2.8</em><strong>4.9</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <b>0.57...</b></span></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><span class="large">cos(35°) </span></td>
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<td><span class="large">= <span class="intbl"><em>Adjacent</em><strong>Hypotenuse</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl"><em>4.0</em><strong>4.9</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <b>0.82...</b></span></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><span class="large">tan(35°) </span></td>
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<td><span class="large">= <span class="intbl"><em>Opposite</em><strong>Adjacent</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl"><em>2.8</em><strong>4.0</strong></span></span></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <b>0.70...</b></span></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Size Does Not Matter</h2>
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<p>The triangle can be large or small and the <b>ratio of sides stays the same</b>.</p>
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<p>Only the angle changes the ratio.</p>
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<p>Try dragging point "A" to change the angle and point "B" to change the size:</p>
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<iframe src="algebra/isincostan.html" scrolling="no" style="width:362px; height:302px; overflow:hidden; margin:auto; display:block; border: none;"></iframe>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="algebra/images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" height="75" width="118"></p>
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<p>Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.</p>
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<p>But you still need to remember <b>what they mean</b>!</p>
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<p>In picture form:</p>
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<p class="center"><img src="algebra/images/sin-cos-tan-divide.svg" alt="sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent individual illustration" style="max-width:100%" height="103" width="609"></p>
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<h3>Practice Here:</h3>
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<iframe src="geometry/itriangle-q.html" scrolling="no" style="width:402px; height:362px; overflow:hidden; margin:auto; display:block; border: none;"></iframe>
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<h2>Sohcahtoa</h2>
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<div class="center80">
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<p>How to remember? Think <b>"Sohcahtoa"</b>!</p>
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<p>It works like this:</p>
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<table align="center" width="379" border="0">
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<tbody>
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<tr>
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<td style="width:88px;">
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<div class="center"><b><i>Soh...</i></b></div></td>
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<td style="width:281px;">
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<div class="center"><b>S</b>ine = <b>O</b>pposite / <b>H</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td style="width:88px;">
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<div class="center"><b><i>...cah...</i></b></div></td>
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<td style="width:281px;">
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<div class="center"><b>C</b>osine = <b>A</b>djacent / <b>H</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td style="width:88px;">
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<div class="center"><b><i>...toa</i></b></div></td>
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<td style="width:281px;">
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<div class="center"><b>T</b>angent = <b>O</b>pposite / <b>A</b>djacent</div></td>
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</tr>
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</tbody></table>
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<p>You can read more about <a href="algebra/sohcahtoa.html">sohcahtoa</a> ... please remember it, it may help in an exam !</p>
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</div>
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<h2>Angles From 0° to 360°</h2>
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<p>Move the mouse around to see how different angles (in <a href="geometry/radians.html">radians</a> or <a href="geometry/degrees.html">degrees</a>) affect sine, cosine and tangent.</p>
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<div class="script" style="height: 560px;">
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algebra/images/circle-triangle.js
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</div>
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<p>In this animation the hypotenuse is 1, making the <a href="geometry/unit-circle.html"> Unit Circle</a>.</p>
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<p>Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also.</p>
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<div style="clear:both"></div>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><i><img src="images/style/smile.svg" alt="smiley" style="float:left; margin: 10px;" height="48" width="48">"Why didn't <b>sin</b> and <b>tan</b><br> go to the party?"<br>
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"... just <b>cos</b>!"</i></td>
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</tr>
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</tbody></table>
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<h2>Examples</h2>
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<div class="example">
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<h3 class="large">Example: what are the sine, cosine and tangent of 30° ?</h3>
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<p>The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of
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√<span style="text-decoration: overline;">3</span>:</p>
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<p class="center"><img src="images/triangle-30.gif" alt="30 degree triangle" height="216" width="298"></p>
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<p>Now we know the lengths, we can calculate the functions:</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td>
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<div class="center"><b>Sine</b></div></td>
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<td style="width:20px;"> </td>
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<td>sin(30°) = 1 / 2 = 0.5</td>
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</tr>
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<tr>
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<td>
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<div class="center"><b>Cosine</b></div></td>
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<td> </td>
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<td>cos(30°) = 1.732 / 2 = 0.866...</td>
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</tr>
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<tr>
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<td>
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<div class="center"><b>Tangent</b></div></td>
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<td> </td>
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<td>tan(30°) = 1 / 1.732 = 0.577...</td>
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</tr>
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</tbody></table>
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<p>(get your calculator out and check them!)</p>
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</div>
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<div class="example">
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<h3 class="large">Example: what are the sine, cosine and tangent of 45° ?</h3>
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<p>The classic 45° triangle has two sides of 1 and a hypotenuse of √<span style="text-decoration: overline;">2</span>:</p>
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<p class="center"><img src="images/triangle-45.gif" alt="45 degree triangle" height="211" width="210"></p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td>
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<div class="center"><b>Sine</b></div></td>
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<td style="width:20px;"> </td>
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<td>sin(45°) = 1 / 1.414 = 0.707...</td>
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</tr>
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<tr>
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<td>
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<div class="center"><b>Cosine</b></div></td>
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<td> </td>
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<td>cos(45°) = 1 / 1.414 = 0.707...</td>
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</tr>
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<tr>
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<td>
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<div class="center"><b>Tangent</b></div></td>
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<td> </td>
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<td>tan(45°) = 1 / 1 = 1</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Why?</h2>
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<p>Why are these functions important?</p>
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<ul>
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<li>Because they let us work out angles when we know sides</li>
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<li>And they let us work out sides when we know angles</li>
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</ul>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="algebra/images/trig-2example2.gif" alt="trig example" height="170" width="242"></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 cable makes a <b>39° angle</b> with the seabed</li>
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<li>The cable has a <b>30 meter length</b>.</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 class="center larger">The depth "d" is <b>18.88 m</b></p>
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</div>
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<h2>Exercise</h2>
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<p>Try this <a href="sine-graph-exercise.html">paper-based exercise</a> where you can calculate the sine function
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for all angles from 0° to 360°, and then graph the result. It will help you to understand these relatively
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simple functions.</p>
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<p>You can also see <a href="algebra/trig-sin-cos-tan-graphs.html">Graphs of Sine, Cosine and Tangent</a>.</p>
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<p>And play with a <a href="physics/spring.html">spring that makes a sine wave</a>.</p>
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<h2>Less Common Functions</h2>
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<p>To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.</p>
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<p>They are equal to <b>1 divided by cos</b>, <b>1 divided by sin</b>, and <b>1 divided by tan</b>:</p>
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<table align="center" cellpadding="2" border="0">
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<tbody>
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<tr>
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<td>
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<div align="right">Secant Function:</div></td>
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<td> </td>
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<td nowrap="nowrap"><b>sec(<i>θ</i>) = <span class="intbl"><em>Hypotenuse</em><strong>Adjacent</strong></span></b></td>
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<td> </td>
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<td><i>(=1/cos)</i></td>
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</tr>
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<tr>
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<td>
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<div align="right">Cosecant Function:</div></td>
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<td> </td>
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<td nowrap="nowrap"><b>csc(<i>θ</i>) = <span class="intbl"><em>Hypotenuse</em><strong>Opposite</strong></span></b></td>
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<td> </td>
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<td><i>(=1/sin)</i></td>
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</tr>
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<tr>
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<td>
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<div align="right">Cotangent Function:</div></td>
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<td> </td>
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<td nowrap="nowrap"><b>cot(<i>θ</i>) = <span class="intbl"><em>Adjacent</em><strong>Opposite</strong></span></b></td>
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<td> </td>
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<td><i>(=1/tan)</i></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="questions">1494, 1495, 724, 725, 1492, 1493, 726, 727, 2362, 2363</div>
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<div class="related">
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<a href="algebra/trig-solving-triangles.html">Solving Triangles</a>
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