{"id":242,"date":"2017-08-23T10:29:38","date_gmt":"2017-08-23T10:29:38","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=242"},"modified":"2017-08-31T04:34:33","modified_gmt":"2017-08-31T04:34:33","slug":"the-unit-circle","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/the-unit-circle\/","title":{"rendered":"The Unit Circle"},"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\/law-of-sines-law-of-cosines\">\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\/introduction-to-tangent-cotangent-secant-and-cosecant\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">The Unit Circle<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will discuss the definition and some applications of the unit circle. We will also learn how\u00a0cosines and sines relate to x- and y-coordinates on a unit circle.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>There are three basic <em><strong>trigonometric ratios\u00a0<\/strong><\/em>that can help us to solve for side lengths and\u00a0angle measures in right triangles.<\/li>\n<li><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p1_clip_image003.gif\" width=\"127\" height=\"44\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p1_clip_image006.gif\" width=\"129\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><span style=\"margin-bottom: 0in;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p1_clip_image009.gif\" width=\"112\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/li>\n<li>A <strong><em>radian<\/em><\/strong> is the length of the\u00a0radius of the circle measured along the circumference of the\u00a0circle, or an arc on the circle with the length of the radius.\u00a0For every 360\u00b0, there are 2<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p1_clip_image012.gif\" width=\"15\" height=\"15\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0radians, and for every 180\u00b0, there are\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p1_clip_image014.gif\" width=\"15\" height=\"15\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0radians.<\/li>\n<\/ul>\n<section>\n<h3>What is the unit circle?<\/h3>\n<p>A <abbr title=\" an easy way to illustrate the two most common angle measurements, degrees and radians, on a circle. The radius of a unit circle is always equal to 1. \">unit circle<\/abbr> is an easy way to illustrate the two types of angle measurement\u00a0that we have learned, degrees and radians, on a circle. A unit\u00a0circle can be described as a circle with a radius of 1 (The radius\u00a0of a unit circle is always equal to 1). <span style=\"text-decoration: none;\">In\u00a0this lesson, we will discuss how to define cosines and sines as <\/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;\">coordinates\u00a0on a unit circle. if you know how to use the unit circle to assist\u00a0you in calculating angles, it can save you a great deal of\u00a0frustration and hard work, especially if you are unable to use a\u00a0calculator to make your calculations. <\/span><\/p>\n<p class=\"lesson_subhead\"><span style=\"text-decoration: none;\">What does a unit ci<\/span>rcle\u00a0look like, and how do sine and cosine relate to it?<\/p>\n<p><span style=\"text-decoration: none;\">A right triangle can be\u00a0placed within the unit circle in order to illustrate how the\u00a0cosine and sine can be used to assist with calculations. The right\u00a0triangle will have an acute angle, the vertex of which is located\u00a0at the center of the circle. The side opposite this angle, which\u00a0we will refer to as <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0will always be perpendicular to the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0and the side adjacent to the angle will lie along the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis.\u00a0This right triangle has legs <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">(adjacent to the angle <\/span><em><span style=\"text-decoration: none;\">\u03b8<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">(opposite to the angle\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p2_clip_image003.gif\" width=\"13\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>)\u00a0and a hypotenuse equal to 1. Remember that the hypotenuse is equal\u00a0to the radius, and we know that the radius of the unit circle is\u00a0always equal to 1.<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20001.JPG\" alt=\" Unit Circle\" width=\"175\" height=\"155\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>The Unit Circle<\/h3>\n<p><span style=\"text-decoration: none;\">Using the trigonometric\u00a0ratios that we learned in a previous section, we know that if we\u00a0were trying to solve for the length of side <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">we could make use of the \u201c-cah\u201d in <\/span><em><span style=\"text-decoration: none;\">soh\u00a0cah toa<\/span><\/em><span style=\"text-decoration: none;\">, because <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the adjacent side, and we know that the length of the\u00a0hypotenuse is equal to 1. So we can use the trigonometric ratio\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image003.gif\" width=\"183\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We now perform the same operation to solve for the length of side\u00a0<\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0We use the \u201csoh-\u201d in <\/span><em><span style=\"text-decoration: none;\">soh\u00a0cah toa<\/span><\/em><span style=\"text-decoration: none;\"> because\u00a0side <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the opposite side of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image006.gif\" width=\"13\" height=\"19\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and we know that the length of the hypotenuse is equal to 1. So we\u00a0can determine that <\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image009.gif\" width=\"184\" height=\"44\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<\/span><\/em><span style=\"text-decoration: none;\">We have now shown, using trigonometric ratios, that cos <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">can be defined by the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-coordinate,\u00a0and sin <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">can be defined by the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-coordinate\u00a0of point <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">on the unit circle. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">The unit circle defined by\u00a0cosine and sine as 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-<\/span><\/em><span style=\"text-decoration: none;\">coordinates\u00a0is shown below. Since the radius of the unit circle is always\u00a0equal to 1, we have the coordinates shown below. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20002.JPG\" alt=\"Unit Circle with coordinates\" width=\"250\" height=\"193\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\">We will attempt a couple of\u00a0examples on the sample unit circle above. Afterwards, you will be\u00a0asked to complete some example problems on your own.<\/p>\n<p><span style=\"text-decoration: none;\">Using the unit circle, can\u00a0you determine why\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image012.gif\" width=\"59\" height=\"16\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>?\u00a0In this case,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image014.gif\" width=\"13\" height=\"19\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0= 0. On the unit circle, we can see that at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image016.gif\" width=\"13\" height=\"19\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0= 0, the coordinates are (1, 0). Since cos\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image018.gif\" width=\"13\" height=\"19\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-coordinate,\u00a0and the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-coordinate is 1, we can determine that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image021.gif\" width=\"57\" height=\"19\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\">We will try one more example\u00a0before you work an example by yourself, to make sure you have a\u00a0good grasp on the unit circle and its applications.<\/p>\n<p><span style=\"text-decoration: none;\">Try to explain why <\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image024.gif\" width=\"72\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><\/em><span style=\"text-decoration: none;\">.\u00a0At\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image026.gif\" width=\"13\" height=\"19\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>=\u00a090\u00b0, the coordinates are (0, 1). Since sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image028.gif\" width=\"13\" height=\"19\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">coordinate,\u00a0and the <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">coordinate\u00a0is equal to 1 at 90\u00b0, we can see that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image030.gif\" width=\"72\" height=\"21\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\n<\/span><\/p>\n<h3>Try and do these examples on your own.<\/h3>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image033.gif\" width=\"17\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li>0<\/li>\n<li>\u20131<\/li>\n<li>None of the above<\/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 A. The answer was given in the problem\u00a0itself. (This is an example where it pays to be able to easily\u00a0convert degrees to radians, and vice versa.). Remember\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image036.gif\" width=\"107\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image039.gif\" width=\"17\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>=\u00a0sin 90\u00b0 = 1.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image042.gif\" width=\"35\" height=\"21\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li>0<\/li>\n<li>\u20131<\/li>\n<li>None of the above<\/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.\u00a0Using the unit circle, you can see that at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image045.gif\" width=\"61\" height=\"21\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0the <\/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;\">coordinates\u00a0are (0, \u20131). Since sin\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image047.gif\" width=\"13\" height=\"19\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">coordinate,\u00a0and the <\/span><em><span style=\"text-decoration: none;\">y-<\/span><\/em><span style=\"text-decoration: none;\">coordinate\u00a0is \u20131, we know that sin 270\u00b0= \u20131. <\/span><\/p>\n<\/div>\n<\/section>\n<p style=\"text-decoration: none;\">This next example is a little\u00a0more difficult. First, we will learn a trick that will make angle\u00a0measures greater than 360 degrees, or 2\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p3_clip_image050.gif\" width=\"15\" height=\"15\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0radians, a little easier, using the unit circle. You may have seen\u00a0this trick before, but we will go over the proper application of\u00a0the trick, just to be sure.<\/p>\n<p><span style=\"text-decoration: none;\">For instance, you want to\u00a0find <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">degrees on a unit circle, where <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is greater than or equal to 360\u00b0. We know that there are 360\u00b0\u00a0in a unit circle, so you can take <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">\u2013 360\u00b0, and keep taking 360\u00b0 off of your answer\u00a0until you get an angle between 0\u00b0 and 360\u00b0. <\/span><\/p>\n<p style=\"text-decoration: none;\">Where is 405\u00b0 on a unit\u00a0circle?<\/p>\n<p style=\"text-decoration: none;\">405\u00b0 \u2013 360\u00b0 = 45\u00b0.\u00a0Therefore, 405\u00b0 on a unit circle is equal to 45\u00b0 on a unit\u00a0circle.<\/p>\n<h3>Where is 800\u00b0 on a unit circle?<\/h3>\n<p>800\u00b0 \u2013 360\u00b0 = 440\u00b0. Since 440\u00b0 is not\u00a0between 0\u00b0 and 360\u00b0, we must repeat this process through\u00a0one more iteration. So, 440\u00b0 \u2013 360\u00b0= 80\u00b0. Since\u00a080\u00b0 is between 0\u00b0 and 360\u00b0, we know that a measurement\u00a0of 800\u00b0 goes all the way around the unit circle twice, and is\u00a0ultimately equivalent to 80\u00b0 on a unit circle.<\/p>\n<p class=\"lesson_subhead\"><strong>For what do you use this trick going forward?<\/strong><\/p>\n<ul>\n<li>The trick will simplify the\u00a0process of determining where a point lies on the unit circle.<\/li>\n<li>If we are trying to find the cosine or sine of a large\u00a0angle, we can use this rule to identify cosines and sines of\u00a0angles we already know. The next example illustrates a fine\u00a0example of this.<\/li>\n<\/ul>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is cos\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image003.gif\" width=\"25\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li>0<\/li>\n<li>\u20131<\/li>\n<li>None of the above<\/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>If you said choice B, then you are correct.<\/p>\n<p>First you need to convert radians to degrees.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image006.gif\" width=\"96\" height=\"44\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image009.gif\" width=\"52\" height=\"44\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image012.gif\" width=\"48\" height=\"21\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now, using the new trick you learned, you can calculate that,<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image015.gif\" width=\"123\" height=\"21\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/strong><\/p>\n<p><span style=\"text-decoration: none;\">Since cos 90\u00b0 points\u00a0to the coordinates (0, 1) and cos is the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-coordinate,\u00a0we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s3_p4_clip_image018.gif\" width=\"87\" height=\"45\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span>.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>A <strong><em>unit circle\u00a0<\/em><\/strong>is an easy way to illustrate the two most common angle\u00a0measurements, degrees and radians, on a circle.<\/li>\n<li>The radius of a unit circle is\u00a0always equal to 1.<\/li>\n<li>You can insert a right triangle into a unit circle, with a\u00a0radius point serving as the hypotenuse, to help you solve for\u00a0sine and cosine by using the trigonometric ratios that we learned\u00a0previously.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/law-of-sines-law-of-cosines\">\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\/introduction-to-tangent-cotangent-secant-and-cosecant\">Next Lesson \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 Lesson \u27a1 The Unit Circle Objective In this lesson, we will discuss the definition and some applications of the unit circle. We will also learn how\u00a0cosines and sines relate to x- and y-coordinates on a unit circle. Previously Covered: There are three basic trigonometric ratios\u00a0that can help us to solve for [&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-242","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/242","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=242"}],"version-history":[{"count":9,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/242\/revisions"}],"predecessor-version":[{"id":590,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/242\/revisions\/590"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=242"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}