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350 lines
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HTML
350 lines
16 KiB
HTML
<!DOCTYPE html>
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<title>Trigonometry</title>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Introduction to Trigonometry</h1>
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<p class="center"><i><b>Trigonometry</b> (from Greek trigonon "triangle" + metron "measure")</i></p>
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<p class="center">Want to learn Trigonometry? Here is a quick summary.<br>
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Follow the links for more, or go to <a href="trigonometry-index.html">Trigonometry Index</a></p>
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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><img src="images/triangle-sss.svg" alt="triangle" height="131" width="220"></td>
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<td><span class="larger"><b>Trigonometry</b> ... is all about <b>triangles.</b> </span></td>
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</tr>
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</tbody></table>
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<p>Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!</p>
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<h2>Right-Angled Triangle</h2>
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<p>The triangle of most interest is the <a href="../right_angle_triangle.html">right-angled triangle</a>. The right angle is shown by the little box in the corner:</p>
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<p class="center"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="190" width="326"></p>
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<p>Another angle is often labeled <span class="large">θ</span>, and the three sides are then called:</p>
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<ul>
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<li><b>Adjacent</b>: adjacent (next to) the angle <span class="large">θ</span></li>
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<li><b>Opposite</b>: opposite the angle <span class="large">θ</span></li>
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<li>and the longest side is the <b>Hypotenuse</b></li>
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</ul><p> </p>
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<div class="fun">
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<h3>Why a Right-Angled Triangle?</h3>
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<p>Why is this triangle so important?</p>
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<p>Imagine we can measure along and up but want to know the direct distance and angle:</p>
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<p class="center"><img src="images/trig-why-dist.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="163" width="189"></p>
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<p>Trigonometry can find that missing angle and distance.</p>
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<p>Or maybe we have a distance and angle and need to "plot the dot" along and up:</p>
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<p class="center"><img src="images/trig-why-screen.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="149" width="238"></p>
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<p>Questions like these are common in engineering, computer animation and more.</p>
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<p>And trigonometry gives the answers!</p>
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</div>
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<h2>Sine, Cosine and Tangent</h2>
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<p>The main functions in trigonometry are <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a></p>
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<p class="center large">They are simply one side of a right-angled triangle divided by another.</p>
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<p>For any angle "<b><i>θ</i></b>":</p>
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<p class="center"><img src="images/sin-cos-tan.svg" alt="sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent" height="181" width="468"></p>
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<p class="center"><i>(Sine, Cosine and Tangent are often abbreviated to <span class="large">sin, cos and tan</span>.)</i></p>
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<p> </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 10px 5px 0;"><img src="../geometry/images/triangle-28-40-49.gif" alt="triangle 2.8 4.0 4.9 has 35 degree angle" 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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<p class="center large">sin(35°) = <span class="intbl"><em>Opposite</em><strong>Hypotenuse</strong></span> = <span class="intbl"><em>2.8</em><strong>4.9</strong></span> = <b>0.57...</b></p>
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<div style="clear:both"></div>
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</div>
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<p>The triangle could be larger, smaller or turned around, but <b>that angle will always have that ratio</b>.</p>
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<p>Calculators have sin, cos and tan to help us, so let's see how to use them:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"> </p> <span style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-45-sin.svg" alt="right angle triangle 45 degrees, hypotenuse 20" height="138" width="185"></span>
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<h3>Example: How Tall is The Tree?</h3>
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<p>We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):</p>
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<ul>
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<li>We know the <b>Hypotenuse</b></li>
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<li>And we want to know the <b>Opposite</b></li>
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</ul>
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<p><b>Sine</b> is the ratio of <b>Opposite / Hypotenuse</b>:</p>
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<p class="center large">sin(45°) = <span class="intbl">
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<em>Opposite</em>
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<strong>Hypotenuse</strong>
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</span></p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" height="75" width="118"></p>
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<p>Get a calculator, type in "45", then the "sin" key:</p>
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<p class="center large">sin(45°) = <b>0.7071...</b></p>
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<p> </p>
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<p class="def">What does the <b>0.7071...</b> mean? It is the ratio of the side lengths, so the Opposite is <i>about 0.7071</i> times as long as the Hypotenuse.</p>
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<p> </p>
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<p>We can now put <b>0.7071...</b> in place of sin(45°):</p>
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<p class="center large"><b>0.7071...</b> = <span class="intbl"> <em>Opposite</em> <strong>Hypotenuse</strong> </span></p>
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<p>And we also know the hypotenuse is <b>20</b>:</p>
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<p class="center large">0.7071... = <span class="intbl"> <em>Opposite</em> <strong><b>20</b></strong></span></p>
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<p>To solve, first multiply both sides by 20:</p>
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<p class="center large">20 × 0.7071... = Opposite</p>
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<p>Finally:</p>
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<p class="center large">Opposite = <b>14.14m</b> (to 2 decimals)</p>
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</div>
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When you gain more experience you can do it quickly like this:
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-45-sin.svg" alt="right angle triangle 45 degrees, hypotenuse 20" height="138" width="185"></p>
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<h3>Example: How Tall is The Tree?</h3>
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<div style="clear:both">
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</div>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">sin(45°) = <span class="intbl">
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<em>Opposite</em>
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<strong>Hypotenuse</strong>
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</span></span></div>
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<div class="row"><span class="left">We know:</span><span class="right">0.7071... = <span class="intbl">
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<em>Opposite</em>
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<strong>20</strong>
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</span></span></div>
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<div class="row"><span class="left">Swap sides:</span><span class="right"><span class="intbl">
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<em>Opposite</em>
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<strong>20</strong>
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</span> = 0.7071...</span></div>
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<div class="row"><span class="left">Multiply both sides by <b>20</b>: </span><span class="right">Opposite = 0.7071... × 20</span></div>
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<div class="row"><span class="left">Calculate:</span><span class="right"><span class="large">Opposite = <b>14.14</b><br>
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(to 2 decimals)</span></span></div>
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</div>
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<p class="larger">The tree is 14.14m tall</p>
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</div>
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<h2>Try Sin Cos and Tan</h2>
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<p>Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.</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>Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be <b>positive or negative</b> by the rules of <a href="../data/cartesian-coordinates.html">Cartesian coordinates</a>, so the sine, cosine and tangent change between positive and negative also.</p>
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<p>So <b>trigonometry is also about <a href="../geometry/circle.html">circles</a></b>!</p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/unit-circle.svg" alt="unit circle" height="222" width="242"></p>
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<h2>Unit Circle</h2>
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<p>What you just played with is the <a href="../geometry/unit-circle.html">Unit Circle</a>.</p>
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<p>It is a circle with a radius of 1 with its center at 0.</p>
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<p>Because the radius is 1, we can directly measure sine, cosine and tangent.</p>
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<p>Here we see the sine function being made by the unit circle:</p>
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<div class="script" style="height: 230px;">
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images/circle-sine.js
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</div>
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<p><i>Note: you can see the nice <a href="trig-sin-cos-tan-graphs.html">graphs made by sine, cosine and tangent</a>.</i></p>
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<h2>Degrees and Radians</h2>
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<p>Angles can be in <a href="../geometry/degrees.html">Degrees</a> or <a href="../geometry/radians.html">Radians</a>. Here are some examples:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th width="200">Angle</th>
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<th width="100">Degrees</th>
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<th width="100">Radians</th>
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</tr>
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<tr style="text-align:center;">
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<td style="width:200px;"><img src="../geometry/images/symbol-right-angle.gif" alt="right angle" height="28" width="28">Right Angle </td>
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<td style="width:100px;">90°</td>
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<td style="width:100px;"><span class="times">π</span>/2</td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:200px;">__ Straight Angle</td>
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<td style="width:100px;">180°</td>
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<td style="width:100px;"><span class="times">π</span></td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:200px;"><img src="../geometry/images/symbol-circle.gif" alt="right angle" height="28" width="28"> Full Rotation</td>
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<td style="width:100px;">360°</td>
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<td style="width:100px;">2<span class="times">π</span></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Repeating Pattern</h2>
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<p>Because the angle is <b>rotating around and around the circle</b> the Sine, Cosine and Tangent functions <b>repeat once every full rotation</b> (see <a href="amplitude-period-frequency-phase-shift.html">Amplitude, Period, Phase Shift and Frequency</a>).</p>
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<p class="center"><img src="images/cosine-repeating.svg" alt="cosine repeates every 360 degrees" height="156" width="404"></p>
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<p>When we want to calculate the function for an angle larger than a full rotation of 360° (2<span class="times">π</span> radians) we subtract as many full rotations as needed to bring it back below 360° (2<span class="times">π</span> radians):</p>
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<div class="example">
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<h3>Example: what is the cosine of 370°?</h3>
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<p>370° is greater than 360° so let us subtract 360°</p>
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<p class="center">370° − 360° = 10°</p>
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<p class="larger">cos(370°) = cos(10°) = <b>0.985</b> (to 3 decimal places)</p>
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</div>
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<p>And when the angle is less than zero, just add full rotations.</p>
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<div class="example">
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<h3>Example: what is the sine of −3 radians?</h3>
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<p>−3 is less than 0<span class="times"></span> so let us add 2<span class="times">π</span> radians</p>
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<p class="center">−3 + 2<span class="times">π</span> = −3 + 6.283... = 3.283... rad<span class="times"></span>ians</p>
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<p class="larger">sin(−3) = sin(3.283...) = <b>−0.141</b> (to 3 decimal places)</p>
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</div>
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<h2>Solving Triangles</h2>
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<p>Trigonometry is also useful for general triangles, not just right-angled ones .</p>
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<p>It helps us in <a href="trig-solving-triangles.html">Solving Triangles</a>. "Solving" means finding missing sides and angles.</p>
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<div class="example">
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<h3>Example: Find the Missing Angle "C"</h3>
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<p class="center"><img src="images/trig-asaex1.gif" alt="trig ASA example" height="125" width="175"></p>
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<p>Angle <b>C</b> can be found using <a href="../proof180deg.html">angles of a triangle add to 180°</a>:</p>
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<p class="center larger">So C = 180° − 76° − 34° = <b>70°</b></p>
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</div>
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<p>We can also find missing side lengths. The general rule is:</p>
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<p class="center"><b>When we know any 3 of the sides or angles we can find the other 3</b><br>
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(except for the three angles case)</p>
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<p class="center large">See <a href="trig-solving-triangles.html">Solving Triangles</a> for more details.</p>
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<h2>Other Functions (Cotangent, Secant, Cosecant)</h2>
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<p>Similar to Sine, Cosine and Tangent, there are three other <b>trigonometric functions</b> which are made by dividing one side by another:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="190" width="326"></p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td>
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||
<div align="right">Cosecant Function:</div>
|
||
</td>
|
||
<td><b>csc(<i>θ</i>) = Hypotenuse / Opposite</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<div align="right">Secant Function:</div>
|
||
</td>
|
||
<td><b>sec(<i>θ</i>) = Hypotenuse / Adjacent</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<div align="right">Cotangent Function:</div>
|
||
</td>
|
||
<td><b>cot(<i>θ</i>) = Adjacent / Opposite</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Trigonometric and Triangle Identities</h2>
|
||
|
||
<p>And as you get better at Trigonometry you can learn these:</p>
|
||
|
||
<table align="center" width="80%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:center;"><img src="../geometry/images/triangle-abc.svg" alt="right angled triangle" height="109" width="189"></td>
|
||
<td>
|
||
<p>The <a href="trigonometric-identities.html">Trigonometric Identities</a> are equations that are true for all <b>right-angled triangles</b>.</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><img src="images/triangle-sss.svg" alt="triangle" height="131" width="220"></td>
|
||
<td>
|
||
<p>The <a href="triangle-identities.html">Triangle Identities</a> are equations that are true for all triangles (they don't have to have a right angle).</p>
|
||
</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<p class="center">Enjoy becoming a triangle (and circle) expert!</p>
|
||
<p class="center"> </p>
|
||
|
||
<div class="related">
|
||
<a href="trigonometry-index.html">Trigonometry Index</a>
|
||
<a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a>
|
||
<a href="../geometry/unit-circle.html">Unit Circle</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
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