{"id":1006,"date":"2024-04-10T14:14:48","date_gmt":"2024-04-10T14:14:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1006"},"modified":"2025-08-17T15:53:15","modified_gmt":"2025-08-17T15:53:15","slug":"more-basic-functions-and-graphs-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/more-basic-functions-and-graphs-background-youll-need-1\/","title":{"raw":"More Basic Functions and Graphs: Background You\u2019ll Need 1","rendered":"More Basic Functions and Graphs: Background You\u2019ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Evaluate trigonometric functions using the unit circle<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Trigonometric Functions and the Unit Circle<\/h2>\r\n<p>When evaluating trigonometric functions, the unit circle is an invaluable tool. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a right triangle, where the hypotenuse is the radius of the circle, and the [latex]x[\/latex] and [latex]y[\/latex] coordinates of the point represent the lengths of the other two sides.<\/p>\r\n\r\n[caption id=\"attachment_12625\" align=\"aligncenter\" width=\"800\"]<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img class=\"wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a> Figure 1. Unit Circle[\/caption]\r\n\r\n<p>The unit circle provides the sine and cosine values for any given angle measure. For each angle, the [latex]x[\/latex]-coordinate represents its cosine value, and the [latex]y[\/latex]-coordinate stands for its sine value. The tangent of an angle is the ratio of the sine to the cosine:<\/p>\r\n<center>[latex]\\tan(\\theta)=\\frac{\\sin(\\theta)}{\\cos(\\theta)}=\\frac{y}{x}[\/latex]<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>unit circle<\/h3>\r\n<p>The <strong>unit circle<\/strong> allows us to evaluate trigonometric functions by using the coordinates of points on the circle.<\/p>\r\n<ul>\r\n\t<li>The [latex]x[\/latex]-coordinate gives the cosine value<\/li>\r\n\t<li>The [latex]y[\/latex]-coordinate gives the sine value<\/li>\r\n\t<li>The ratio of [latex]y[\/latex] to [latex]x[\/latex] gives the tangent value for any given angle.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function\u2019s value at a given angle.<\/p>\r\n<\/section>\r\n<p>Certain angles have coordinates that can be easily remembered:<\/p>\r\n<ul>\r\n\t<li>At [latex]0[\/latex] degrees (or [latex]0[\/latex] radians), the coordinates are ([latex]1, 0[\/latex]), so [latex]\\cos(0) = 1[\/latex] and [latex]\\sin(0) = 0[\/latex].<\/li>\r\n\t<li>At [latex]90[\/latex] degrees (or [latex]\\frac{\\pi}{2}[\/latex] radians), the coordinates are ([latex]0, 1[\/latex]), so [latex]\\cos\\left(\\frac{\\pi}{2}\\right) = 0[\/latex] and [latex]\\sin\\left(\\frac{\\pi}{2}\\right) = 1[\/latex].<\/li>\r\n\t<li>At [latex]180[\/latex] degrees (or [latex]\\pi[\/latex] radians), the coordinates are ([latex]-1, 0[\/latex]), so [latex]\\cos(\\pi) = -1[\/latex] and [latex]\\sin(\\pi) = 0[\/latex].<\/li>\r\n\t<li>At [latex]270[\/latex] degrees (or [latex]\\frac{3\\pi}{2}[\/latex] radians), the coordinates are ([latex]0, -1[\/latex]), so [latex]\\cos\\left(\\frac{3\\pi}{2}\\right) = 0[\/latex] and [latex]\\sin\\left(\\frac{3\\pi}{2}\\right) = -1[\/latex].<\/li>\r\n<\/ul>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Evaluate at Any Angle using the Unit Circle<\/strong><\/p>\r\n<p>For any angle [latex]\\theta[\/latex], you can determine its corresponding point on the unit circle by:<\/p>\r\n<ol>\r\n\t<li>Start with your given angle [latex]\\theta[\/latex]. Position it so that it starts at the positive [latex]x[\/latex]-axis and opens counterclockwise for positive angles, or clockwise for negative angles.<\/li>\r\n\t<li>Extend the angle's terminal side until it intersects the unit circle at a point [latex]P[\/latex].<\/li>\r\n\t<li>The coordinates of point [latex]P(x,y)[\/latex] on the unit circle give you the cosine and sine of [latex]\\theta[\/latex] respectively.\r\n\r\n<ul>\r\n\t<li>[latex]\\cos(\\theta)[\/latex] is the [latex]x[\/latex]-coordinate of point [latex]P[\/latex].<\/li>\r\n\t<li>[latex]\\sin(\\theta)[\/latex] is the [latex]y[\/latex]-coordinate of point [latex]P[\/latex].\u00a0<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Remember that the signs of the sine and cosine are determined by the quadrant in which point [latex]P[\/latex]\u00a0lies:\r\n\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>In Quadrant I, both sine and cosine are positive.<\/li>\r\n\t<li>In Quadrant II, sine is positive and cosine is negative.<\/li>\r\n\t<li>In Quadrant III, both sine and cosine are negative.<\/li>\r\n\t<li>In Quadrant IV, sine is negative and cosine is positive.\u00a0<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p>The terminal side of an angle is the side that moves or rotates from the initial side to form the angle. The position of the terminal side after this rotation determines the magnitude of the angle.<\/p>\r\n<center>\r\n[caption id=\"attachment_1044\" align=\"alignnone\" width=\"469\"]<img class=\"wp-image-1044 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525.png\" alt=\"Illustration of an angle with labels for initial side, terminal side, and vertex.\" width=\"469\" height=\"263\" \/> Diagram showing the vertex, initial side, and terminal side of an angle[\/caption]\r\n<\/center><\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"913342\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"913342\"]<\/p>\r\n<p>[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]173155-16024[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Evaluate trigonometric functions using the unit circle<\/li>\n<\/ul>\n<\/section>\n<h2>Trigonometric Functions and the Unit Circle<\/h2>\n<p>When evaluating trigonometric functions, the unit circle is an invaluable tool. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a right triangle, where the hypotenuse is the radius of the circle, and the [latex]x[\/latex] and [latex]y[\/latex] coordinates of the point represent the lengths of the other two sides.<\/p>\n<figure id=\"attachment_12625\" aria-describedby=\"caption-attachment-12625\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a><figcaption id=\"caption-attachment-12625\" class=\"wp-caption-text\">Figure 1. Unit Circle<\/figcaption><\/figure>\n<p>The unit circle provides the sine and cosine values for any given angle measure. For each angle, the [latex]x[\/latex]-coordinate represents its cosine value, and the [latex]y[\/latex]-coordinate stands for its sine value. The tangent of an angle is the ratio of the sine to the cosine:<\/p>\n<div style=\"text-align: center;\">[latex]\\tan(\\theta)=\\frac{\\sin(\\theta)}{\\cos(\\theta)}=\\frac{y}{x}[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3>unit circle<\/h3>\n<p>The <strong>unit circle<\/strong> allows us to evaluate trigonometric functions by using the coordinates of points on the circle.<\/p>\n<ul>\n<li>The [latex]x[\/latex]-coordinate gives the cosine value<\/li>\n<li>The [latex]y[\/latex]-coordinate gives the sine value<\/li>\n<li>The ratio of [latex]y[\/latex] to [latex]x[\/latex] gives the tangent value for any given angle.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Remember, every angle in quadrant two, three, or four has a reference angle that lies in quadrant one. The quadrant of the original angle only affects the sign (positive or negative) of a trigonometric function\u2019s value at a given angle.<\/p>\n<\/section>\n<p>Certain angles have coordinates that can be easily remembered:<\/p>\n<ul>\n<li>At [latex]0[\/latex] degrees (or [latex]0[\/latex] radians), the coordinates are ([latex]1, 0[\/latex]), so [latex]\\cos(0) = 1[\/latex] and [latex]\\sin(0) = 0[\/latex].<\/li>\n<li>At [latex]90[\/latex] degrees (or [latex]\\frac{\\pi}{2}[\/latex] radians), the coordinates are ([latex]0, 1[\/latex]), so [latex]\\cos\\left(\\frac{\\pi}{2}\\right) = 0[\/latex] and [latex]\\sin\\left(\\frac{\\pi}{2}\\right) = 1[\/latex].<\/li>\n<li>At [latex]180[\/latex] degrees (or [latex]\\pi[\/latex] radians), the coordinates are ([latex]-1, 0[\/latex]), so [latex]\\cos(\\pi) = -1[\/latex] and [latex]\\sin(\\pi) = 0[\/latex].<\/li>\n<li>At [latex]270[\/latex] degrees (or [latex]\\frac{3\\pi}{2}[\/latex] radians), the coordinates are ([latex]0, -1[\/latex]), so [latex]\\cos\\left(\\frac{3\\pi}{2}\\right) = 0[\/latex] and [latex]\\sin\\left(\\frac{3\\pi}{2}\\right) = -1[\/latex].<\/li>\n<\/ul>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Evaluate at Any Angle using the Unit Circle<\/strong><\/p>\n<p>For any angle [latex]\\theta[\/latex], you can determine its corresponding point on the unit circle by:<\/p>\n<ol>\n<li>Start with your given angle [latex]\\theta[\/latex]. Position it so that it starts at the positive [latex]x[\/latex]-axis and opens counterclockwise for positive angles, or clockwise for negative angles.<\/li>\n<li>Extend the angle&#8217;s terminal side until it intersects the unit circle at a point [latex]P[\/latex].<\/li>\n<li>The coordinates of point [latex]P(x,y)[\/latex] on the unit circle give you the cosine and sine of [latex]\\theta[\/latex] respectively.\n<ul>\n<li>[latex]\\cos(\\theta)[\/latex] is the [latex]x[\/latex]-coordinate of point [latex]P[\/latex].<\/li>\n<li>[latex]\\sin(\\theta)[\/latex] is the [latex]y[\/latex]-coordinate of point [latex]P[\/latex].\u00a0<\/li>\n<\/ul>\n<\/li>\n<li>Remember that the signs of the sine and cosine are determined by the quadrant in which point [latex]P[\/latex]\u00a0lies:\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>In Quadrant I, both sine and cosine are positive.<\/li>\n<li>In Quadrant II, sine is positive and cosine is negative.<\/li>\n<li>In Quadrant III, both sine and cosine are negative.<\/li>\n<li>In Quadrant IV, sine is negative and cosine is positive.\u00a0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox recall\">\n<p>The terminal side of an angle is the side that moves or rotates from the initial side to form the angle. The position of the terminal side after this rotation determines the magnitude of the angle.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_1044\" aria-describedby=\"caption-attachment-1044\" style=\"width: 469px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1044 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525.png\" alt=\"Illustration of an angle with labels for initial side, terminal side, and vertex.\" width=\"469\" height=\"263\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525.png 469w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525-300x168.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525-65x36.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525-225x126.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10163538\/Screenshot-2024-04-10-123525-350x196.png 350w\" sizes=\"(max-width: 469px) 100vw, 469px\" \/><figcaption id=\"caption-attachment-1044\" class=\"wp-caption-text\">Diagram showing the vertex, initial side, and terminal side of an angle<\/figcaption><\/figure>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the coordinates of the point on the unit circle at an angle of [latex]\\frac{5\\pi }{3}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q913342\">Show Solution<\/button><\/p>\n<div id=\"q913342\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm173155\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173155-16024&theme=lumen&iframe_resize_id=ohm173155&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1006"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1006\/revisions"}],"predecessor-version":[{"id":4737,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1006\/revisions\/4737"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1006\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1006"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1006"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1006"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}