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469 lines
19 KiB
HTML
469 lines
19 KiB
HTML
<!DOCTYPE html>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Sine, Cosine and Tangent in Four Quadrants</h1>
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<h2>Sine, Cosine and Tangent</h2>
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<p>The three main functions in trigonometry are <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a>.</p>
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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" height="190" width="326"></p>
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<p>They are easy to calculate:</p>
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<p class="center"><b>Divide the length of one side of a<br>
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right angled triangle by another side</b></p>
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<p class="center"><br>
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... but we must know which sides!</p>
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<div style="clear:both"></div>
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<p align="left">For an angle <b><i>θ</i></b>, the functions are calculated this way:</p>
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<div class="simple">
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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">Sine Function: </div></td>
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<td><b>sin(<i>θ</i>) = Opposite / Hypotenuse</b></td>
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</tr>
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<tr>
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<td>
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<div align="right">Cosine Function: </div></td>
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<td><b>cos(<i>θ</i>) = Adjacent / Hypotenuse</b></td>
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</tr>
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<tr>
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<td>
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<div align="right">Tangent Function: </div></td>
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<td><b>tan(<i>θ</i>) = Opposite / Adjacent</b></td>
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</tr>
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</tbody></table><br>
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</div>
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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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<tbody>
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<tr>
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<td><img src="../geometry/images/triangle-28-40-49.gif" alt="triangle 2.8 4.0 4.9" height="117" width="159"></td>
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<td>
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<p>Using this triangle (lengths are only to one decimal place):</p>
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<p class="larger">sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = <b>0.57...</b></p></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Cartesian Coordinates</h2>
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<p>Using <a href="../data/cartesian-coordinates.html">Cartesian Coordinates</a> we mark a point on a graph by <b>how far along</b> and <b>how far up</b> it is:</p>
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<p class="center"><img src="../geometry/images/coordinates-cartesian.svg" alt="graph with point (12,5)" height="232" width="348"><br>
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<span class="larger">The point <b>(12,5)</b> is 12 units along, and 5 units up.</span></p><p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/cartesian-quadrants.svg" alt="Quadrants" height="191" width="250"></p>
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<h2>Four Quadrants</h2>
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<p>When we include <b>negative values</b>, the x and y axes divide the space up into 4 pieces:</p>
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<p class="center"><b>Quadrants I, II, III</b> and<b> IV</b></p>
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<p><i>(They are numbered in a counter-clockwise direction)</i></p>
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<ul>
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<li>In <b>Quadrant I</b> both x and y are positive,</li>
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<li>in <b>Quadrant II</b> <span style="color: #ff0000;">x is negative</span> (y is still positive),</li>
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<li>in <b>Quadrant III</b> <span style="color: #ff0000;">both x and y are negative</span>, and</li>
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<li>in <b>Quadrant IV</b> x is positive again, and <span style="color: #ff0000;">y is negative</span>.</li>
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</ul>
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<p>Like this:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/trig-quadrants-signs.svg" alt="Quadrant Signs" height="276" width="276"></p>
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<div class="beach" style="min-width:300px;">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th>Quadrant</th>
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<th>X<br>
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(horizontal)</th>
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<th>Y<br>
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(vertical)</th>
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<th>Example</th>
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</tr>
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<tr style="text-align:center;">
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<td class="larger"><b>I</b></td>
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<td>Positive</td>
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<td>Positive</td>
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<td>(3,2)</td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger"><b>II</b></td>
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<td><i>Negative</i></td>
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<td>Positive</td>
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<td> (−5,4)</td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger"><b>III</b></td>
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<td><i>Negative</i></td>
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<td><i>Negative</i></td>
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<td>(−2,−1)</td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger"><b>IV</b></td>
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<td>Positive</td>
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<td><i>Negative</i></td>
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<td> (4,−3)</td>
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</tr>
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</tbody></table>
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</div>
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<div style="clear:both"></div>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../data/images/cartesian-coordinates.gif" alt="cartesian coordinates" height="286" width="284"></p>
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<p>Example: The point "C" (−2,−1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).</p>
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<p>Both x and y are negative, so that point is in "Quadrant III"</p>
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</div>
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<h2>Reference Angle</h2>
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<p>Angles can be more than 90º</p>
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<p>But we can bring them back below 90º using the x-axis as the reference.</p>
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<p class="center"><i>Think "reference" means "refer x"</i></p>
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<p>The simplest method is to do a sketch!</p>
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<div class="example">
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<h3>Example: 160º</h3>
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<p>Start at the positive x axis and rotate 160º</p>
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<p class="center"><img src="images/trig-quadrant2-ref-ex.svg" alt="triangle quadrant example" height="167" width="251"><br>
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Then find the angle to the nearest part of the x-axis,<br>
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in this case 20º</p><br>
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<p>The reference angle for 160º is <b>20º</b></p>
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</div>
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<p>Here we see four examples with a reference angle of 30º:</p>
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<p class="center"><img src="images/trig-reference-30.svg" alt="30 degree reference angles" height="208" width="301"></p>
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<p>Instead of a sketch you can use these rules:</p>
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td style="text-align: center;">Quadrant</td>
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<td style="text-align: center;">Reference Angle</td>
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</tr>
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<tr>
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<td style="text-align: center;">I</td>
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<td style="text-align: center;">θ</td></tr>
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<tr>
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<td style="text-align: center;">II</td>
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<td style="text-align: center;">180º − θ</td>
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</tr>
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<tr>
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<td style="text-align: center;">III</td>
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<td style="text-align: center;">θ − 180º</td>
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</tr>
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<tr>
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<td style="text-align: center;">IV</td>
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<td style="text-align: center;">360º − θ</td>
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</tr>
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</tbody>
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</table>
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<h2>Sine, Cosine and Tangent in the
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Four Quadrants</h2>
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<p>Now let us look at the details of a <b>30° right triangle</b> in each of the 4 Quadrants.</p>
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<p>In <span class="larger">Quadrant I</span> everything is normal, and <a href="../sine-cosine-tangent.html">Sine, Cosine and Tangent</a> are all positive:</p>
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<div class="example">
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<h3>Example: The sine, cosine and tangent of 30°</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/trig-quadrant1-ex.svg" alt="triangle 30 quadrant I" height="171" width="249"></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 nowrap="nowrap">
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<div class="center">sin(30°) = 1 / 2 = 0.5</div></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 nowrap="nowrap">
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<div class="center">cos(30°) = 1.732 / 2 = 0.866</div></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 nowrap="nowrap">
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<div class="center">tan(30°) = 1 / 1.732 = 0.577</div></td>
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</tr>
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</tbody></table>
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<div style="clear:both"></div>
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</div>
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<p> </p>
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<p>But in <span class="larger">Quadrant II</span>, the <b>x direction is negative</b>, and cosine and tangent become negative:</p>
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<div class="example">
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<h3>Example: The sine, cosine and tangent of 150°</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/trig-quadrant2-ex.svg" alt="triangle 30 quadrant I" height="171" width="249"></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 nowrap="nowrap">
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<div class="center">sin(150°) = 1 / 2 = 0.5</div></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 nowrap="nowrap">
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<div class="center">cos(150°) = <span class="hilite">−1.732</span> / 2 = <span class="hilite">−0.866</span></div></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 nowrap="nowrap">
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<div class="center">tan(150°) = 1 / <span class="hilite">−1.732</span> = <span class="hilite">−0.577</span></div></td>
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</tr>
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</tbody></table>
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<div style="clear:both"></div>
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</div>
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<p> </p>
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<p><span style="font-weight: bold;"></span>In <span class="larger">Quadrant III</span>, sine and cosine are negative:</p>
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<div class="example">
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<h3>Example: The sine, cosine and tangent of 210°</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/trig-quadrant3-ex.svg" alt="triangle 30 quadrant I" height="171" width="249"></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 nowrap="nowrap">
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<div class="center">sin(210°) = <span class="hilite">−1</span> / 2 = <span class="hilite">−0.5</span></div></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 nowrap="nowrap">
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<div class="center">cos(210°) = <span class="hilite">−1.732</span> / 2 = <span class="hilite">−0.866</span></div></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 nowrap="nowrap">
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<div class="center">tan(210°) = <span class="hilite">−1 / −1.732</span> = 0.577</div></td>
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</tr>
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</tbody></table>
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<p>Note: Tangent is <b>positive</b> because dividing a negative by a negative gives a positive.</p>
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<div style="clear:both"></div>
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</div>
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<p> </p>
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<p><span style="font-weight: bold;"></span>In <span class="larger">Quadrant IV</span>, sine and tangent are negative:</p>
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<div class="example">
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<h3>Example: The sine, cosine and tangent of 330°</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/trig-quadrant4-ex.svg" alt="triangle 30 quadrant I" height="171" width="249"></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 nowrap="nowrap">
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<div class="center">sin(330°) = <span class="hilite">−1</span> / 2 = <span class="hilite">−0.5</span></div></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>
|
||
<td nowrap="nowrap">
|
||
<div class="center">cos(330°) = 1.732 / 2 = 0.866</div></td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<div class="center"><b>Tangent</b></div></td>
|
||
<td nowrap="nowrap">
|
||
<div class="center">tan(330°) = <span class="hilite">−1</span> / 1.732 = <span class="hilite">−0.577</span></div></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div style="clear:both"></div>
|
||
</div>
|
||
<p><span class="large">There is a pattern!</span> Look at when Sine Cosine and Tangent are <b>positive</b> ...</p>
|
||
<ul>
|
||
<li><span style="font-weight: bold;">All</span> three of them are positive
|
||
in <b>Quadrant I</b></li>
|
||
<li><span style="font-weight: bold;">Sine</span> only is positive in <b>Quadrant II</b></li>
|
||
<li><span style="font-weight: bold;">Tangent</span> only is positive in <b>Quadrant III</b></li>
|
||
<li><span style="font-weight: bold;">Cosine</span> only is positive in <b>Quadrant IV</b></li>
|
||
</ul>
|
||
<p>This can be shown even easier by:</p>
|
||
<p class="center"><img src="images/trig-quadrants-astc.svg" alt="trig ASTC is All,Sine,Tangent,Cosine" height="166" width="261"></p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-graph-quadrants.svg" alt="trig graph 4 quadrants" height="210" width="271"><br>
|
||
This graph shows "ASTC" also.</p>
|
||
<p>Some people like to remember the four letters <span style="font-weight: bold;">ASTC</span> by one of these:</p>
|
||
<ul>
|
||
<li><span style="font-weight: bold;">A</span>ll <span style="font-weight: bold;">S</span>tudents <span style="font-weight: bold;">T</span>ake <span style="font-weight: bold;">C</span>hemistry</li>
|
||
<li><span style="font-weight: bold;">A</span>ll <span style="font-weight: bold;">S</span>tudents <span style="font-weight: bold;">T</span>ake <span style="font-weight: bold;">C</span>alculus</li>
|
||
<li><span style="font-weight: bold;">A</span>ll <span style="font-weight: bold;">S</span>illy <span style="font-weight: bold;">T</span>om <span style="font-weight: bold;">C</span>ats</li>
|
||
<li><span style="font-weight: bold;">A</span>ll <span style="font-weight: bold;">S</span>tations <span style="font-weight: bold;">T</span>o <span style="font-weight: bold;">C</span>entral</li>
|
||
<li><b>A</b>dd <b>S</b>ugar <b>T</b>o <b>C</b>offee</li>
|
||
</ul>
|
||
<p>Maybe you could make up one of your own. Or
|
||
just remember <span style="font-weight: bold;">ASTC</span>.</p>
|
||
|
||
|
||
<h2>Inverse Sin, Cos and Tan</h2>
|
||
|
||
<p>What is the <a href="trig-inverse-sin-cos-tan.html">Inverse Sine</a> of 0.5? </p>
|
||
<p class="center larger">sin<sup>-1</sup>(0.5) = ?</p>
|
||
<p>In other words, when y is 0.5 on the graph below, what is the angle?</p>
|
||
<p class="center"><img src="images/trig-inverse-sin-cos-tan2.svg" alt="sine crosses 0.5 at 30,150,390, etc" style="max-width:100%" height="264" width="585"><span class="larger"><br>
|
||
There are <b>many angles</b> where y=0.5<br>
|
||
</span></p>
|
||
<p align="left">The trouble is: <b>a calculator will only give you one of those values</b> ...</p>
|
||
<p class="center" align="left">... but there are always two values between 0º and 360º<br>
|
||
(and infinitely many beyond):</p>
|
||
|
||
<table align="center" cellspacing="2" cellpadding="2" border="1">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align: center;"><br>
|
||
</td>
|
||
<td style="text-align: center;">First value</td>
|
||
<td style="text-align: center;">Second value</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align: center;">Sine</td>
|
||
<td style="text-align: center;">θ</td>
|
||
<td style="text-align: center;">180º − θ</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align: center;">Cosine</td>
|
||
<td style="text-align: center;">θ</td>
|
||
<td style="text-align: center;">360º − θ</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align: center;">Tangent</td>
|
||
<td style="text-align: center;">θ</td>
|
||
<td style="text-align: center;">θ + 180º</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p>We can now solve equations for
|
||
any angle!</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve sin θ = 0.5</h3>
|
||
<p>We get the first solution from the calculator = sin<sup>-1</sup>(0.5) = 30º
|
||
(it is in Quadrant I)</p>
|
||
<p>The next solution is 180º − 30º = 150º (Quadrant II)</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve cos θ
|
||
= −0.85</h3>
|
||
<p>We get the first solution from the calculator = cos<sup>-1</sup>(−0.85) =
|
||
148.2º (Quadrant II)</p>
|
||
<p>The other solution is 360º − 148.2º = 211.8º (Quadrant III)</p>
|
||
</div>
|
||
<p>We may need to bring our angle between 0º and 360º by adding or subtracting 360º</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve tan θ
|
||
= −1.3</h3>
|
||
<p>We get the first solution from the calculator = tan<sup>-1</sup>(−1.3) = −52.4º</p>
|
||
<p>This is less than 0º, so we add 360º: −52.4º + 360º = 307.6º (Quadrant IV)</p>
|
||
<p>The other solution is −52.4º + 180º =
|
||
127.6º (Quadrant II)</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">3914, 3915, 3916, 3917, 3918, 3919, 3920, 3921, 3922, 3923</div>
|
||
<div class="activity"> <a href="../activity/walk-in-desert-2.html">Activity: A Walk in the Desert 2</a> </div>
|
||
|
||
<div class="related">
|
||
<a href="../geometry/unit-circle.html">Unit Circle</a>
|
||
<a href="trig-interactive-unit-circle.html">Interactive Unit Circle</a>
|
||
<a href="trigonometry-index.html">Trigonometry Index</a>
|
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</div>
|
||
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