{"id":249,"date":"2017-08-23T10:32:41","date_gmt":"2017-08-23T10:32:41","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=249"},"modified":"2017-10-12T09:43:45","modified_gmt":"2017-10-12T09:43:45","slug":"rectangular-and-polar-coordinates","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/rectangular-and-polar-coordinates\/","title":{"rendered":"Rectangular and Polar Coordinates"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/half-angle-and-double-angle-formulas\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Next Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Rectangular and Polar Coordinates<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will define rectangular and polar coordinates and discuss the differences between and how to\u00a0convert between these two different types of coordinates.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p1_clip_image003.gif\" width=\"127\" height=\"44\" align=\"absmiddle\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p1_clip_image006.gif\" width=\"129\" height=\"44\" align=\"absmiddle\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p1_clip_image009.gif\" width=\"112\" height=\"44\" align=\"absmiddle\" \/><\/li>\n<\/ul>\n<section>\n<h3><\/h3>\n<h3>What are rectangular coordinates?<\/h3>\n<p><em><span style=\"text-decoration: none;\">Rectangular coordinates\u00a0<\/span><\/em><span style=\"text-decoration: none;\">a<\/span>re coordinates stated in the form <span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">)<\/span><em><span style=\"text-decoration: none;\">.\u00a0<\/span><\/em><span style=\"text-decoration: none;\">Y<\/span>ou are probably already familiar with rectangular\u00a0coordinates from previous experiences. You may have used\u00a0rectangular coordinates in the past without realizing what they\u00a0were called. the point (3, 3), given in rectangular coordinates,\u00a0is graphed below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/newgraph.gif\" alt=\"new graph\" width=\"189\" height=\"183\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">As you can see, this point\u00a0is 3 units to the right of the origin along the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0and 3 units up from the origin along the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis.<br \/>\n<\/span><\/p>\n<h4>What are polar coordinates?<\/h4>\n<p><abbr title=\"coordinates used to plot points and figures in the form (d, c), where d is the distance from the origin to the point, and c is the angle between the x-axis and the point \">Polar\u00a0coordinates<\/abbr> are <span style=\"text-decoration: none;\">stated in\u00a0the form (<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p2_clip_image003.gif\" width=\"13\" height=\"19\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>),\u00a0where <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the distance from the origin to the point, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p2_clip_image005.gif\" width=\"13\" height=\"19\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the angle between the positive <\/span><em><span style=\"text-decoration: none;\">x-<\/span><\/em><span style=\"text-decoration: none;\">axis\u00a0and the ray from the origin to the point. An example of the polar\u00a0coordinates (<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p2_clip_image007.gif\" width=\"13\" height=\"19\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>)\u00a0is shown below. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.4%20Art%20002.JPG\" alt=\"Polar coordinates (r, theta)\" width=\"175\" height=\"180\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Pay close attention to which form is being used!<\/p>\n<p><span style=\"text-decoration: none;\">In both forms, an ordered\u00a0pair denotes a point. It is impossible to determine which system\u00a0is being used if you are simply given an ordered pair, but not\u00a0told which type of coordinate it is. You may think that a point\u00a0given as (2, <\/span><em><span style=\"text-decoration: none;\">\u03c0<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0must be in polar coordinates, since it contains what appears to be\u00a0a radian measure of an angle, but (2, <\/span><em><span style=\"text-decoration: none;\">\u03c0<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0is also a perfectly valid point in the rectangular coordinate\u00a0system. A well stated problem will always let you know which\u00a0system is being used. <\/span><\/p>\n<h3>How do you convert from polar to rectangular coordinates?<\/h3>\n<p>You can convert from polar to rectangular coordinates using the\u00a0trigonometric ratios that you learned earlier and the diagram of\u00a0polar coordinates shown below.<\/p>\n<p>A right triangle can be inserted into the diagram. This step\u00a0may look familiar because it is very similar to the process we used in the section on the unit circle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.4%20Art%20003.JPG\" alt=\" Diagram of polar coordinates\" width=\"175\" height=\"184\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">The legs of the right\u00a0triangle are designated <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">because they represent the <\/span><em><span style=\"text-decoration: none;\">x-\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y-\u00a0<\/span><\/em><span style=\"text-decoration: none;\">axes. We want <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">to be stated in terms of polar coordinates (<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image003.gif\" width=\"13\" height=\"19\" name=\"graphics4\" align=\"TEXTTOP\" border=\"0\" \/>).\u00a0Using the trigonometric ratios, we know that <\/span><strong><span style=\"text-decoration: none;\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image006.gif\" width=\"152\" height=\"47\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/span><\/strong><span style=\"text-decoration: none;\">When\u00a0solving for <\/span><em><span style=\"text-decoration: none;\">y,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">you get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image009.gif\" width=\"68\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p><span style=\"text-decoration: none;\">We also know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image012.gif\" width=\"155\" height=\"47\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0When solving for <\/span><em><span style=\"text-decoration: none;\">x,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">you get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image015.gif\" width=\"69\" height=\"19\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p>From this process, we can conclude that the equation for\u00a0converting polar coordinates to rectangular coordinates is:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image018.gif\" width=\"156\" height=\"21\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice correctly converts polar coordinates\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image021.gif\" width=\"55\" height=\"45\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0to rectangular coordinates?<\/p>\n<ol>\n<li>(2, 2<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image024.gif\" width=\"24\" height=\"24\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<li>(3,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image026.gif\" width=\"24\" height=\"24\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<li>(2,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image029.gif\" width=\"16\" height=\"41\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<li>(1,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image032.gif\" width=\"27\" height=\"45\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is A.\u00a0We know <\/span><em><span style=\"text-decoration: none;\">r <\/span><\/em><span style=\"text-decoration: none;\">=\u00a04 and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image035.gif\" width=\"44\" height=\"41\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image038.gif\" width=\"156\" height=\"21\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so using substitution, you get <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image041.gif\" width=\"193\" height=\"51\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>,<\/p>\n<p style=\"text-decoration: none;\">and<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image044.gif\" width=\"216\" height=\"53\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">So converting polar\u00a0coordinates\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image047.gif\" width=\"55\" height=\"45\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0into rectangular coordinates equals\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p3_clip_image050.gif\" width=\"57\" height=\"25\" name=\"graphics20\" align=\"absmiddle\" \/><\/span>.<\/p>\n<\/div>\n<\/section>\n<h4>How do you convert from\u00a0rectangular to polar coordinates?<\/h4>\n<p class=\"lesson_subhead\" style=\"text-decoration: none;\" align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/rectangularcoordinatesxy.gif\" alt=\"Rectangular coordinates (x,y)\" width=\"189\" height=\"183\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Using the diagram of the\u00a0rectangular coordinates (<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">) above, you can\u00a0see that a right triangle can be inserted in this graph also. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Again, we have legs <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and since we are trying to convert from rectangular coordinates to\u00a0polar coordinates, we need to solve for <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image003.gif\" width=\"13\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in terms of <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\">We know by the Pythagorean\u00a0Theorem, that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image006.gif\" width=\"80\" height=\"25\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p style=\"text-decoration: none;\">Take the square root of both\u00a0sides to get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image009.gif\" width=\"87\" height=\"31\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Now that we have solved fo<span style=\"text-decoration: none;\">r\u00a0<\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in terms of <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">we need to solve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image011.gif\" width=\"13\" height=\"19\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in terms of <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p>By the trigonometric ratio, we know that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image014.gif\" width=\"141\" height=\"44\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Solving for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image016.gif\" width=\"13\" height=\"19\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0by using our inverse trigonometric rule, we know that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image019.gif\" width=\"91\" height=\"45\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>From this process, we can conclude that the equation that\u00a0converts rectangular coordinates to polar coordinates is:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image037.gif\" width=\"208\" height=\"53\" align=\"absmiddle\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows rectangular coordinates (2, 2) converted to\u00a0polar coordinates?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image025.gif\" width=\"64\" height=\"45\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image028.gif\" width=\"60\" height=\"26\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image031.gif\" width=\"67\" height=\"45\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image034.gif\" width=\"55\" height=\"45\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is C.\u00a0We know that <\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">=\u00a02 , <\/span><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">=\u00a02, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image037.gif\" width=\"208\" height=\"53\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So using substitution you get <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image040.gif\" width=\"179\" height=\"29\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>,<\/p>\n<p style=\"text-decoration: none;\">and<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image043.gif\" width=\"159\" height=\"45\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>So, converting rectangular coordinates\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image046.gif\" width=\"43\" height=\"24\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0into polar coordinates equals\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p4_clip_image049.gif\" width=\"67\" height=\"45\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Rectangular and Polar Coordinates<\/h3>\n<p>Not only can you convert between polar and rectangular\u00a0coordinates, but you can also convert a polar equation into a\u00a0rectangular equation or vice versa by using substitution.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the equ<span style=\"text-decoration: none;\">ation\u00a0<\/span><em><span style=\"text-decoration: none;\">y = <\/span><\/em><span style=\"text-decoration: none;\">3<\/span><em><span style=\"text-decoration: none;\">x\u00a0+ <\/span><\/em><span style=\"text-decoration: none;\">2 in polar form?<br \/>\n<\/span><\/p>\n<ol>\n<li><em><span style=\"text-decoration: none;\">r = <\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image003.gif\" width=\"95\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/li>\n<li><em><span style=\"text-decoration: none;\">r = <\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image006.gif\" width=\"16\" height=\"41\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/li>\n<li><em><span style=\"text-decoration: none;\">r = <\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image009.gif\" width=\"93\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/li>\n<li><em><span style=\"text-decoration: none;\">r = <\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image012.gif\" width=\"49\" height=\"41\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is C.<\/p>\n<p>Since we know\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image015.gif\" width=\"68\" height=\"21\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image018.gif\" width=\"69\" height=\"19\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we can use substitution to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image021.gif\" width=\"141\" height=\"29\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image024.gif\" width=\"129\" height=\"19\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image027.gif\" width=\"129\" height=\"19\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image030.gif\" width=\"133\" height=\"29\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image033.gif\" width=\"115\" height=\"44\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<p class=\"lesson_subhead\">How do you write complex numbers in polar form?<\/p>\n<p><abbr title=\" is the field of numbers in the form a + bi where a and b are real numbers and i is an imaginary number equal to . \">Complex\u00a0numbers<\/abbr> are numbers which contain both a real part and an\u00a0imaginary part. Remember, imaginary <span style=\"text-decoration: none;\">numbers\u00a0and complex numbers are not the same thing. Imaginary numbers are\u00a0scalar multiples of <\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where<\/span><em><span style=\"text-decoration: none;\"> i <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p5_clip_image036.gif\" width=\"35\" height=\"23\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Some examples of imaginary numbers are 5<\/span><em><span style=\"text-decoration: none;\">i,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">\u20136.2<\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and \u2013<\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0Complex numbers have two parts, a real part and an imaginary part,\u00a0and are written as the sum of these two parts. Some examples of\u00a0complex numbers are 3 + 5<\/span><em><span style=\"text-decoration: none;\">i\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and 298 \u2013 9.3<\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0We traditionally use <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">as\u00a0variables representing real numbers. For complex numbers, the\u00a0letter <\/span><em><span style=\"text-decoration: none;\">z <\/span><\/em><span style=\"text-decoration: none;\">is\u00a0commonly used. Below are plots that include complex numbers. A\u00a0complex number can be thought of as <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ <\/span><em><span style=\"text-decoration: none;\">yi<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where <\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">is\u00a0the real part of <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">yi <\/span><\/em><span style=\"text-decoration: none;\">is\u00a0the imaginary part. To graph a single real number, we need only\u00a0one axis. To graph a single complex number, we need two axes\u2014one\u00a0<\/span>to keep track of the real part of our number, and one to keep\u00a0track of the complex part. We measure the real part of the number\u00a0along the <em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0and the imaginary part along the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis.\u00a0The graph below on the right illustrates how the complex number <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 3 + 3<\/span><em><span style=\"text-decoration: none;\">i\u00a0<\/span><\/em><span style=\"text-decoration: none;\">can be graphed in this way. The same complex number can be denoted\u00a0using polar coordinates as well. The graph on the left illustrates\u00a0how we can begin thinking about writing complex numbers in this\u00a0form. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/newgraph2.gif\" alt=\"newgraph\" width=\"506\" height=\"199\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Take a minute and use the information from this lesson to\u00a0determine how to write complex number in polar form.<\/p>\n<p>You know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image003.gif\" width=\"156\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>So, you have\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image006.gif\" width=\"41\" height=\"20\" name=\"graphics4\" align=\"TEXTTOP\" border=\"0\" \/>.\u00a0Using substitution, you get<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image009.gif\" width=\"269\" height=\"49\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<p>This is the formula for complex numbers in the polar form.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the pola<span style=\"text-decoration: none;\">r form of\u00a0the complex number <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 5 \u2013 5<\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">?<br \/>\n<\/span><\/p>\n<ol>\n<li>(<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image012.gif\" width=\"28\" height=\"22\" name=\"graphics6\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image015.gif\" width=\"25\" height=\"44\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<li><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image018.gif\" width=\"32\" height=\"25\" name=\"graphics8\" align=\"TEXTTOP\" border=\"0\" \/>(cos\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image021.gif\" width=\"25\" height=\"44\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0+ <\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image024.gif\" width=\"25\" height=\"44\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>)<br \/>\n<\/span><\/li>\n<li>(25,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image027.gif\" width=\"25\" height=\"44\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>)<\/li>\n<li><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 25(cos\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image030.gif\" width=\"25\" height=\"44\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0+ <\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image033.gif\" width=\"25\" height=\"44\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>)<br \/>\n<\/span><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p style=\"text-decoration: none;\">The correct choice is B.<\/p>\n<p align=\"CENTER\"><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 5 \u2013 5<\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><\/p>\n<p><span style=\"text-decoration: none;\">This point will be in the\u00a04th quadrant of the complex plane, and since the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-coordinate\u00a0and <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">coordinate<br \/>\nare both equidistant from the origin, the angle must be\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image036.gif\" width=\"25\" height=\"44\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We also know that <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image039.gif\" width=\"63\" height=\"31\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so <\/span><em><span style=\"text-decoration: none;\">r <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image042.gif\" width=\"76\" height=\"25\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So the polar form of the complex number is therefore <\/span><em><span style=\"text-decoration: none;\">z\u00a0<\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image045.gif\" width=\"32\" height=\"25\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>(cos\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image048.gif\" width=\"25\" height=\"44\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0+ <\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image051.gif\" width=\"25\" height=\"44\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>).<br \/>\n<\/span><\/p>\n<\/div>\n<\/section>\n<p>This method can be used to multiply complex numbers in the\u00a0polar form. Just remember to use substitution when doing this. Try\u00a0the example below.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>W<span style=\"text-decoration: none;\">hat is <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= (3<\/span><em><span style=\"text-decoration: none;\">i\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ 2)(6<\/span><em><span style=\"text-decoration: none;\">i\u00a0\u2013 <\/span><\/em><span style=\"text-decoration: none;\">2) in\u00a0polar form? <\/span><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image054.gif\" width=\"60\" height=\"44\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image057.gif\" width=\"76\" height=\"44\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image060.gif\" width=\"107\" height=\"44\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image066.gif\" width=\"76\" height=\"44\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D.<\/p>\n<p align=\"CENTER\"><em><span style=\"text-decoration: none;\">y <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0(3<\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">+\u00a02)(6<\/span><em><span style=\"text-decoration: none;\">i <\/span><\/em><span style=\"text-decoration: none;\">\u2013\u00a02) <\/span><\/p>\n<p style=\"text-decoration: none;\">Multiplying out this expression\u00a0you get<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image069.gif\" width=\"140\" height=\"21\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">The \u201318 comes from\u00a03<\/span><em><span style=\"text-decoration: none;\">i\u00a0<\/span><\/em><span style=\"text-decoration: none;\">being multiplied by 6<\/span><em><span style=\"text-decoration: none;\">i,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">because\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image072.gif\" width=\"112\" height=\"29\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image075.gif\" width=\"75\" height=\"21\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\">We know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image078.gif\" width=\"69\" height=\"19\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image081.gif\" width=\"68\" height=\"21\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So using substitution we get<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image084.gif\" width=\"105\" height=\"19\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p6_clip_image087.gif\" width=\"76\" height=\"44\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>What is DeMoivre\u2019s Theorem?<\/h3>\n<p>An important theorem that involves complex numbers is\u00a0<em><span style=\"text-decoration: none;\">DeMoivre\u2019s Theorem<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0<\/span><abbr title=\"states that , and can be useful when trying to find the powers of complex numbers and when simplifying complex numbers.\">DeMoivre\u2019s\u00a0Theorem<\/abbr> states that if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image012.gif\" alt=\"l\" width=\"150\" height=\"21\" name=\"graphics3\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image006.gif\" width=\"179\" height=\"29\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Try and prove this theorem yourself. The easiest way of doing this\u00a0is to establish a pattern by starting with\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image009.gif\" width=\"19\" height=\"20\" name=\"graphics5\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0then add a power each time you simplify the equation, until you\u00a0feel comfortable with the theorem. DeMoivre\u2019s Theorem is\u00a0useful when trying to find the powers of complex numbers and to\u00a0simplify complex numbers.<\/p>\n<p>DeMoivre\u2019s Theorem can also relate to addition and double\u00a0angle formulas. To practice working this type of problem, try the\u00a0example below.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>If\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image012.gif\" width=\"150\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0wher<span style=\"text-decoration: none;\">e <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 4, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image015.gif\" width=\"13\" height=\"19\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0= 60\u00b0, what is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image017.gif\" width=\"19\" height=\"20\" name=\"graphics8\" align=\"TEXTTOP\" border=\"0\" \/>?<br \/>\n<\/span><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image020.gif\" width=\"32\" height=\"19\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image023.gif\" width=\"52\" height=\"24\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image026.gif\" width=\"16\" height=\"41\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image029.gif\" width=\"59\" height=\"24\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>T<span style=\"text-decoration: none;\">he correct choice is D.\u00a0By DeMoivre\u2019s Theorem, we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image032.gif\" width=\"179\" height=\"29\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0And since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s9_p7_html_m430f2110.gif\" width=\"37\" height=\"17\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image034.gif\" width=\"13\" height=\"19\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0= 60\u00b0, we can use substitution to get <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image037.gif\" width=\"224\" height=\"31\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image040.gif\" width=\"207\" height=\"31\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p style=\"text-decoration: none;\">Using the double angle formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image043.gif\" width=\"140\" height=\"24\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we know that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s9_p7_html_7eb1f791.gif\" width=\"79\" height=\"29\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/>\u00a0=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image046.gif\" width=\"295\" height=\"56\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0And using the double angle formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image049.gif\" width=\"143\" height=\"21\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we find that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image052.gif\" width=\"320\" height=\"48\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So using substitution, we get<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image055.gif\" width=\"113\" height=\"53\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p7_clip_image058.gif\" width=\"71\" height=\"24\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li><em><strong>Rectangular\u00a0coordinates<\/strong><\/em> are coor<span style=\"text-decoration: none;\">dinates\u00a0written in the form (<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">)<\/span><em><span style=\"text-decoration: none;\">.<\/span><\/em><span style=\"text-decoration: none;\"><br \/>\n<\/span><\/li>\n<li><strong><em>Polar coordinates\u00a0<\/em><\/strong>are written in <span style=\"text-decoration: none;\">the form (<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p8_clip_image003.gif\" width=\"13\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>),\u00a0where <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the distance from the origin to the point, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p8_clip_image005.gif\" width=\"13\" height=\"19\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the angle between the positive <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0ray from the origin to the point. <\/span><\/li>\n<li>The formula to convert from polar coordinates to\u00a0rectangular coordinates is<\/li>\n<\/ul>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p8_clip_image008.gif\" width=\"156\" height=\"21\" name=\"graphics5\" align=\"MIDDLE\" border=\"0\" \/><\/p>\n<ul>\n<li>The formula to convert from rectangular coordinates to\u00a0polar coordinates is<\/li>\n<\/ul>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p8_clip_image011.gif\" width=\"208\" height=\"53\" name=\"graphics6\" align=\"MIDDLE\" border=\"0\" \/><\/p>\n<ul>\n<li><strong><em>Complex numbers\u00a0<\/em><\/strong>are numbers that have both a real-number part and an\u00a0imaginary-number part. Imaginary numbers are scalar multip<span style=\"text-decoration: none;\">les\u00a0of <\/span><em><span style=\"text-decoration: none;\">i<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0where<\/span><em><span style=\"text-decoration: none;\"> i <\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s9_p8_clip_image014.gif\" width=\"35\" height=\"23\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/li>\n<li><strong><em>DeMoivre\u2019s Theorem<\/em><\/strong> states\u00a0that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s9_p8_html_1d7f3e2.gif\" width=\"191\" height=\"27\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and can be useful when trying to find the powers of complex\u00a0numbers, and when simplifying complex numbers.<\/li>\n<\/ul>\n<h3>Further Reading in Trigonometry<\/h3>\n<p><em>Geometry and Trigonometry for Calculus<\/em>. (Peter H. Selby):\u00a0John Wiley and Sons, 1976.<\/p>\n<p><em>Precalculus<\/em>. (Ron Larson): Houghton Mifflin, 2003.<\/p>\n<p><em>Trigonometry<\/em>. (I.M. Gelfand and M. Saul): Springer\u00a0Verlag, 2001.<\/p>\n<p><em>Trigonometry Demystified<\/em>. (Stan Gibilisco): McGraw-Hill,\u00a02003.<\/p>\n<p align=\"CENTER\"><strong><em>Don&#8217;t forget to test your knowledge\u00a0with the <a href=\"http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/trigonometry\" target=\"popsome\"> Trigonometry Chapter Quiz; <\/a><\/em><\/strong><\/p>\n<\/section>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/half-angle-and-double-angle-formulas\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/probability-statistics-data-analysis\">Next Workshop \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Workshop \u27a1 Rectangular and Polar Coordinates Objective In this lesson, we will define rectangular and polar coordinates and discuss the differences between and how to\u00a0convert between these two different types of coordinates. Previously Covered: What are rectangular coordinates? Rectangular coordinates\u00a0are coordinates stated in the form (x,\u00a0y).\u00a0You are probably already familiar with [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-249","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/249","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=249"}],"version-history":[{"count":14,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/249\/revisions"}],"predecessor-version":[{"id":837,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/249\/revisions\/837"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}