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356 lines
14 KiB
HTML
356 lines
14 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Polar and Cartesian Coordinates</h1>
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<p class="center"><b>... and how to convert between them.</b></p>
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<p><i>In a hurry? Read the <a href="#summary">Summary</a>. But please read why first:</i></p>
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<p>To pinpoint where we are on a map or graph there are two main systems:</p>
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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 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="coordinates cartesian (12,5)" height="232" width="348"></p>
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<h2>Polar Coordinates</h2>
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<p>Using Polar Coordinates we mark a point by <b>how far away</b>, and <b>what angle</b> it is:</p>
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<p class="center"><img src="geometry/images/coordinates-polar.svg" alt="coordinates polar 13 at 22.6 degrees" height="232" width="348"></p>
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<h2>Converting</h2>
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<p>To convert from one to the other we will use this triangle:</p>
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<div class="center"><img src="images/coordinates-triangle.gif" alt="coordinates triangle" height="180" width="310"> </div><br>
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<h2>To Convert from Cartesian to Polar</h2>
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<p>When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,<i>θ</i>) we <b>solve a right triangle with two known sides</b>.</p>
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<h3 class="larger">Example: What is (12,5) in Polar Coordinates?</h3>
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<p class="center"><img src="images/coordinates-to-polar.gif" alt="coordinates to polar" height="180" width="310"></p>
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<p>Use <a href="pythagoras.html">Pythagoras Theorem</a> to find the long side (the hypotenuse):</p>
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<div class="so">r<sup>2</sup> = 12<sup>2</sup> + 5<sup>2</sup></div>
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<div class="so">r = √ (12<sup>2</sup> + 5<sup>2</sup>)</div>
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<div class="so">r = √ (144 + 25)
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</div>
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<div class="so">r = √ (169)
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= <b>13</b></div>
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<p>Use the <a href="sine-cosine-tangent.html">Tangent Function</a> to find the angle:</p>
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<div class="so">tan( <i>θ</i> ) = 5 / 12</div>
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<div class="so"><i>θ</i> = tan<sup>-1 </sup>( 5 / 12 ) = <b>22.6°</b> (to one decimal)</div>
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<p class="larger"><b>Answer</b>: the point (12,5) is <b>(13, 22.6°)</b> in Polar Coordinates.</p>
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<div class="def">
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<img src="algebra/images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" style="float:right; margin: 10px;" height="75" width="118">
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<h3>What is <b>tan<sup>-1</sup></b> ?</h3>
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<p class="larger">It is the <a href="algebra/trig-inverse-sin-cos-tan.html">Inverse Tangent Function</a>:</p>
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<ul>
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<li><b class="larger">Tangent</b> takes an angle and gives us a ratio,</li>
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<li><b class="larger">Inverse Tangent</b> takes a ratio (like "5/12") and gives us an angle.</li>
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</ul>
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</div> <p> </p>
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<h3><b>Summary</b>: to convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):</h3>
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<ul>
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<div class="bigul">
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<li><b align="center">r = √ ( x<sup>2</sup> + y<sup>2 </sup>)</b></li>
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<li><b align="center"><i>θ</i> = tan<sup>-1 </sup>( y / x )</b></li>
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</div>
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</ul>
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<p>Note: Calculators may give the wrong value of <b>tan<sup>-1 </sup>()</b> when x or y are negative ... see below for more.</p>
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<h2>To Convert from Polar to Cartesian</h2>
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<p>When we know a point in Polar Coordinates (r, <i>θ</i>), and we want it in Cartesian Coordinates (x,y) we <b>solve a right triangle with a known long side and angle</b>:</p>
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<div class="example">
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<h3>Example: What is (13, 22.6°) in Cartesian Coordinates?</h3>
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<div class="center"><img src="images/coordinates-to-cartesian.gif" alt="to cartesian coordinates" height="180" width="310"><br>
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<br>
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<table cellspacing="0" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td style="text-align:right;">Use the <a href="sine-cosine-tangent.html">Cosine Function</a> for x:</td>
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<td> </td>
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<td><span class="larger">cos( 22.6° ) = x / 13</span></td>
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</tr>
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<tr>
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<td style="text-align:right;">Rearranging and solving:</td>
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<td> </td>
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<td><span class="larger">x = 13 × cos( 22.6° ) </span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td> </td>
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<td><span class="larger">x = 13 × 0.923 </span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td> </td>
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<td><span class="larger">x = <b>12.002...</b></span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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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 style="text-align:right;">Use the <a href="sine-cosine-tangent.html">Sine Function</a> for y:</td>
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<td> </td>
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<td><span class="larger">sin( 22.6° ) = y / 13</span></td>
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</tr>
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<tr>
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<td style="text-align:right;">Rearranging and solving:</td>
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<td> </td>
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<td><span class="larger">y = 13 × sin( 22.6° ) </span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td> </td>
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<td><span class="larger">y = 13 × 0.391 </span></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td> </td>
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<td><span class="larger">y = <b>4.996...</b></span></td>
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</tr>
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</tbody></table>
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</div>
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<p class="larger">Answer: the point (13, 22.6°) is <i>almost exactly</i> <b>(12, 5)</b> in Cartesian Coordinates.</p>
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</div>
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<h3><b>Summary</b>: to convert from Polar Coordinates (r,<i>θ</i>) to Cartesian Coordinates (x,y) :</h3>
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<ul>
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<div class="bigul">
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<li><b align="center">x = r</b> × <b>cos( <i>θ</i> )</b></li>
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<li><b align="center">y = r</b> × <b>sin(<i> θ</i> )</b></li>
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</div>
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</ul>
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<h3>How to Remember?</h3>
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<div class="center80">
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<p class="center"><b>(x,y) is alphabetical,<br>
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<b>(cos,sin)</b> is also alphabetical</b></p>
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</div>
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<p>Also <i><b>"y and sine rhyme"</b></i> (try saying it!)</p>
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<h2>But What About Negative Values of X and Y?</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="data/images/cartesian-quadrants.gif" alt="Quadrants" height="191" width="250"></p>
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<h3>Four Quadrants</h3>
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<p>When we include negative values, the x and y axes divide the<br>
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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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<p>When converting from <span class="larger">Polar to Cartesian</span> coordinates it all works out nicely:</p>
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<div class="example">
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<h3>Example: What is (12, 195°) in Cartesian coordinates?</h3>
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<p>r = 12 and θ = 195°</p>
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<ul>
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<li>x = 12 × cos(195°)<br>
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x = 12 × −0.9659...<br>
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x = <b>−11.59</b> to 2
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decimal places</li>
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<li>y = 12 × sin(195°)<br>
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y = 12 × −0.2588...<br>
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y = <b>−3.11</b> to 2
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decimal places</li>
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</ul>
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<p>So the point is at<b> (−11.59, −3.11)</b>, which is in Quadrant III</p>
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</div>
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<p>But when converting from <span class="larger">Cartesian to Polar</span> coordinates ...</p>
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<p class="center larger">... the calculator can give the <b>wrong value of tan<sup>-1</sup></b></p>
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<p>It all depends what Quadrant the point is in! Use this to fix things:</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;"><b>Value of tan<sup>-1</sup></b></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;">Use
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the calculator value</td>
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</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;">Add
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180° to the calculator value</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;">Add
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180° to the calculator value</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;">Add
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360° to the calculator value</td>
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</tr>
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</tbody>
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</table><br>
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<div class="example"> <img src="geometry/images/polar-example-1.gif" alt="polar example 1" style="float:right; margin: 10px;" height="168" width="178">
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<h3>Example: P = (−3, 10)</h3>
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<p>P is in <b>Quadrant II</b></p>
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<ul>
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<li>r = √((−3)<sup>2</sup> + 10<sup>2</sup>)<br>
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r = √109 = <b>10.4</b> to 1 decimal place</li>
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<li>θ = tan<sup>-1</sup>(10/−3)<br>
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θ = tan<sup>-1</sup>(−3.33...)</li>
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</ul>
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<p>The calculator value for tan<sup>-1</sup>(−3.33...) is −73.3°</p>
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<div class="so"> The rule for Quadrant II is: <span style="font-weight: bold;">Add
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180° to the calculator value</span></div>
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<div class="so">θ = −73.3° + 180<span style="font-weight: bold;">° =</span> 106.7°</div>
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<p>So the Polar Coordinates for the point (−3, 10) are <b>(10.4, 106.7°)</b></p>
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</div>
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<div class="example"> <img src="geometry/images/polar-example-2.gif" alt="polar example 2" style="float:right; margin: 10px;" height="163" width="181">
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<h3>Example: Q = (5, −8)</h3>
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<p>Q is in <b>Quadrant IV</b></p>
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<ul>
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<li>r = √(5<sup>2</sup> + (−8)<sup>2</sup>)<br>
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r
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= √89
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= <b>9.4</b> to 1 decimal place</li>
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<li>θ = tan<sup>-1</sup>(−8/5)<br>
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θ = tan<sup>-1</sup>(−1.6)</li>
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</ul>
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<p>The calculator value for tan<sup>-1</sup>(−1.6) is −58.0°</p>
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<div class="so">The rule for Quadrant IV is: <span style="font-weight: bold;">Add
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360° to the calculator value</span></div>
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<div class="so">θ = −58.0° + 360<span style="font-weight: bold;">° =</span> 302.0°</div>
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<p>So the Polar Coordinates for the point (5, −8) are <b>(9.4, 302.0°)</b></p>
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</div>
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<p> </p>
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<h2><a id="summary"></a>Summary</h2>
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<div class="dotpoint">
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<p>To convert from Polar Coordinates (r,<i>θ</i>) to Cartesian Coordinates (x,y) :</p>
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<ul>
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<li><b>x = r</b> × <b>cos( <i>θ</i> )</b></li>
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<li><b>y = r</b> × <b>sin(<i> θ</i> )</b></li>
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</ul>
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</div>
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<div class="dotpoint">
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<p class="larger">To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):</p>
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<ul>
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<li><b>r = √ ( x<sup>2</sup> + y<sup>2 </sup>)</b></li>
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<li><b><i>θ</i> = tan<sup>-1 </sup>( y / x )</b></li>
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</ul>
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</div>
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<div class="dotpoint">
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<p>The value of <b>tan<sup>-1</sup>( y/x )</b> may need to be adjusted:</p>
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<ul>
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<li>Quadrant
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I:<span style="text-align: center;"> Use
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the calculator value</span></li>
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<li>Quadrant
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II:<span style="text-align: center;"> Add
|
||
180°</span></li>
|
||
<li>Quadrant
|
||
III: <span style="text-align: center;">Add
|
||
180°</span></li>
|
||
<li>Quadrant
|
||
IV: <span style="text-align: center;">Add
|
||
360°</span></li>
|
||
</ul>
|
||
|
||
</div><p> </p>
|
||
<div class="activity"> <a href="activity/walk-in-desert-2.html">Activity: A Walk in the Desert 2</a> </div>
|
||
<div class="questions">2167, 2168, 2169, 2170, 2171, 2172, 2173, 2174, 5159, 5160</div>
|
||
|
||
<div class="related">
|
||
<a href="data/cartesian-coordinates-interactive.html">Interactive Cartesian Coordinates</a>
|
||
<a href="data/cartesian-coordinates.html">Cartesian Coordinates</a>
|
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<a href="data/graphs-index.html">Graphs Index</a>
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<a href="geometry/index.html">Geometry Index</a>
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