{"id":1035,"date":"2024-04-10T15:54:18","date_gmt":"2024-04-10T15:54:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1035"},"modified":"2024-08-05T12:24:59","modified_gmt":"2024-08-05T12:24:59","slug":"trigonometric-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/trigonometric-functions-learn-it-1\/","title":{"raw":"Trigonometric Functions: Learn It 1","rendered":"Trigonometric Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Switch between degree and radian measurements for angles<\/li>\r\n\t<li>Understand and use the basic rules and relationships in trigonometry<\/li>\r\n\t<li>Analyze trigonometric functions by examining their graphs, identifying cycles, and describing shifts in sine and cosine graphs<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Degrees versus Radians<\/h2>\r\n<p>Probably the most familiar unit of angle measurement is the <strong>degree<\/strong>. One degree is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains [latex]360[\/latex] degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol [latex]\u00b0[\/latex].<\/p>\r\n<section class=\"textbox connectIt\">\r\n<p>You will use degrees a lot as you learn geometry but you may see it in other areas of life as well. Degrees are used in:<\/p>\r\n<ul>\r\n\t<li>Navigation - navigation systems, such as compasses and GPS devices, use degrees to indicate directions.<\/li>\r\n\t<li>Construction - degrees are used in construction and engineering to measure and specify angles when building structures. Architects, carpenters, and engineers use degrees to determine the angle of roof slopes, the inclination of ramps, or the angles of intersecting beams.<\/li>\r\n\t<li>Astronomy - astronomers use degrees to describe the positions of celestial objects, angular separations between stars or planets, and the size of apparent motions of celestial bodies<\/li>\r\n\t<li>Sports - in golf, angles are used to calculate the direction and trajectory of shots. In basketball, the angle of a player's jump shot can affect the ball's path to the basket<\/li>\r\n\t<li>Art and design - when creating perspective drawings or determining the tilt and angles of lines in graphic design, degrees are used to ensure accurate proportions and compositions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p><strong>Radians<\/strong> provide an alternative to degrees for measuring angles and are often preferred in mathematics because they have a natural relationship with circle geometry. Radians are based on the concept of using the radius of a circle to measure angles.<\/p>\r\n<p>One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\u03c0[\/latex] times the radius, a full circular rotation is 2[latex]\u03c0[\/latex] radians. The symbol for radians is [latex]\\text{rad}[\/latex].\u00a0<\/p>\r\n<section class=\"textbox recall\">\r\n<p>Pi ([latex]\u03c0[\/latex]) is a mathematical constant approximately equal to [latex]3.14[\/latex]. It represents the ratio of a circle's circumference to its diameter.<\/p>\r\n<\/section>\r\n<p>Since there are [latex]360[\/latex] degrees in a circle and [latex]2\u03c0[\/latex] radians in a circle, the conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex]. It is expected to keep the [latex]\u03c0[\/latex] symbol when discussing radians, not converting to decimals, in order to maintain precision.<\/p>\r\n<center>[latex]\\begin{array}{rcl} 2\\pi \\text{ radians} &amp; = &amp; 360^\\circ \\\\ \\pi \\text{ radians} &amp; = &amp; \\frac{360^\\circ}{2} = 180^\\circ \\\\ 1 \\text{ radian} &amp; = &amp; \\frac{180^\\circ}{\\pi} \\approx 57.3^\\circ \\end{array}[\/latex]<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>degrees versus radians<\/h3>\r\n<ul>\r\n\t<li><strong>Degrees <\/strong>are the most common measurement of angles. A circle is divided into [latex]360^\\circ[\/latex]. The symbol for degrees is a small, raised circle: [latex]^\\circ[\/latex].<\/li>\r\n\t<li><strong>Radians <\/strong>are an alternative unit of angle measurement. In a circle, there are [latex]2\u03c0[\/latex] radians. The symbol for radians is [latex]\\text{rad}[\/latex].<\/li>\r\n<\/ul>\r\n<p>The conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Degrees are commonly used in everyday situations like navigation, construction, and basic geometry, while radians are more prevalent in advanced mathematics, physics, engineering, and other scientific fields.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<ol id=\"fs-id1170572204405\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Express [latex]225\u00b0[\/latex] using radians.<\/li>\r\n\t<li>Express [latex]\\dfrac{5\\pi}{3}[\/latex] rad using degrees.<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1170572224739\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572224739\"]<\/p>\r\n<p id=\"fs-id1170572224739\">Use the fact that [latex]180\u00b0 [\/latex]is equivalent to [latex]\\pi [\/latex] radians as a conversion factor: [latex]1=\\dfrac{\\pi \\, \\text{rad}}{180^{\\circ}}=\\dfrac{180^{\\circ}}{\\pi \\, \\text{rad}}[\/latex].<\/p>\r\n<ol id=\"fs-id1170572240548\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]225^{\\circ}=225^{\\circ}\u00b7\\dfrac{\\pi }{180^{\\circ}}=\\dfrac{5\\pi }{4}\\text{ rad}[\/latex]<\/li>\r\n\t<li>[latex]\\dfrac{5\\pi }{3}\\text{ rad}[\/latex]= [latex]\\dfrac{5\\pi }{3}\u00b7\\dfrac{180^{\\circ}}{\\pi }=300^{\\circ}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">A Ferris wheel with a radius of [latex]20[\/latex] meters makes one complete rotation every [latex]5[\/latex] minutes. A passenger boards at the bottom of the wheel. After [latex]1[\/latex] minute, what is the angle of rotation in:<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">degrees?<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">radians?<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"68351\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"68351\"]<\/p>\r\n<p>Angle of rotation after [latex]1[\/latex] minute:<\/p>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Full rotation takes [latex]5[\/latex] minutes<\/li>\r\n\t<li class=\"whitespace-normal break-words\">In [latex]1[\/latex] minute, it rotates [latex]\\frac{1}{5}[\/latex] of a full circle<\/li>\r\n<\/ul>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>In degrees: [latex](\\frac{1}{5}) \u00d7 360\u00b0 = 72\u00b0[\/latex]<\/li>\r\n\t<li>In radians: [latex](\\frac{1}{5}) \u00d7 2\u03c0 \\text{ rad} = \\frac{2\u03c0}{5} rad \u2248 1.26 \\text{ rad}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]284090[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Switch between degree and radian measurements for angles<\/li>\n<li>Understand and use the basic rules and relationships in trigonometry<\/li>\n<li>Analyze trigonometric functions by examining their graphs, identifying cycles, and describing shifts in sine and cosine graphs<\/li>\n<\/ul>\n<\/section>\n<h2>Degrees versus Radians<\/h2>\n<p>Probably the most familiar unit of angle measurement is the <strong>degree<\/strong>. One degree is [latex]\\frac{1}{360}[\/latex] of a circular rotation, so a complete circular rotation contains [latex]360[\/latex] degrees. An angle measured in degrees should always include the unit \u201cdegrees\u201d after the number, or include the degree symbol [latex]\u00b0[\/latex].<\/p>\n<section class=\"textbox connectIt\">\n<p>You will use degrees a lot as you learn geometry but you may see it in other areas of life as well. Degrees are used in:<\/p>\n<ul>\n<li>Navigation &#8211; navigation systems, such as compasses and GPS devices, use degrees to indicate directions.<\/li>\n<li>Construction &#8211; degrees are used in construction and engineering to measure and specify angles when building structures. Architects, carpenters, and engineers use degrees to determine the angle of roof slopes, the inclination of ramps, or the angles of intersecting beams.<\/li>\n<li>Astronomy &#8211; astronomers use degrees to describe the positions of celestial objects, angular separations between stars or planets, and the size of apparent motions of celestial bodies<\/li>\n<li>Sports &#8211; in golf, angles are used to calculate the direction and trajectory of shots. In basketball, the angle of a player&#8217;s jump shot can affect the ball&#8217;s path to the basket<\/li>\n<li>Art and design &#8211; when creating perspective drawings or determining the tilt and angles of lines in graphic design, degrees are used to ensure accurate proportions and compositions<\/li>\n<\/ul>\n<\/section>\n<p><strong>Radians<\/strong> provide an alternative to degrees for measuring angles and are often preferred in mathematics because they have a natural relationship with circle geometry. Radians are based on the concept of using the radius of a circle to measure angles.<\/p>\n<p>One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals [latex]2\u03c0[\/latex] times the radius, a full circular rotation is 2[latex]\u03c0[\/latex] radians. The symbol for radians is [latex]\\text{rad}[\/latex].\u00a0<\/p>\n<section class=\"textbox recall\">\n<p>Pi ([latex]\u03c0[\/latex]) is a mathematical constant approximately equal to [latex]3.14[\/latex]. It represents the ratio of a circle&#8217;s circumference to its diameter.<\/p>\n<\/section>\n<p>Since there are [latex]360[\/latex] degrees in a circle and [latex]2\u03c0[\/latex] radians in a circle, the conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex]. It is expected to keep the [latex]\u03c0[\/latex] symbol when discussing radians, not converting to decimals, in order to maintain precision.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} 2\\pi \\text{ radians} & = & 360^\\circ \\\\ \\pi \\text{ radians} & = & \\frac{360^\\circ}{2} = 180^\\circ \\\\ 1 \\text{ radian} & = & \\frac{180^\\circ}{\\pi} \\approx 57.3^\\circ \\end{array}[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>degrees versus radians<\/h3>\n<ul>\n<li><strong>Degrees <\/strong>are the most common measurement of angles. A circle is divided into [latex]360^\\circ[\/latex]. The symbol for degrees is a small, raised circle: [latex]^\\circ[\/latex].<\/li>\n<li><strong>Radians <\/strong>are an alternative unit of angle measurement. In a circle, there are [latex]2\u03c0[\/latex] radians. The symbol for radians is [latex]\\text{rad}[\/latex].<\/li>\n<\/ul>\n<p>The conversion factor between degrees and radians is [latex]\\frac{180}{\u03c0}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Degrees are commonly used in everyday situations like navigation, construction, and basic geometry, while radians are more prevalent in advanced mathematics, physics, engineering, and other scientific fields.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<ol id=\"fs-id1170572204405\" style=\"list-style-type: lower-alpha;\">\n<li>Express [latex]225\u00b0[\/latex] using radians.<\/li>\n<li>Express [latex]\\dfrac{5\\pi}{3}[\/latex] rad using degrees.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572224739\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572224739\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572224739\">Use the fact that [latex]180\u00b0[\/latex]is equivalent to [latex]\\pi[\/latex] radians as a conversion factor: [latex]1=\\dfrac{\\pi \\, \\text{rad}}{180^{\\circ}}=\\dfrac{180^{\\circ}}{\\pi \\, \\text{rad}}[\/latex].<\/p>\n<ol id=\"fs-id1170572240548\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]225^{\\circ}=225^{\\circ}\u00b7\\dfrac{\\pi }{180^{\\circ}}=\\dfrac{5\\pi }{4}\\text{ rad}[\/latex]<\/li>\n<li>[latex]\\dfrac{5\\pi }{3}\\text{ rad}[\/latex]= [latex]\\dfrac{5\\pi }{3}\u00b7\\dfrac{180^{\\circ}}{\\pi }=300^{\\circ}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">A Ferris wheel with a radius of [latex]20[\/latex] meters makes one complete rotation every [latex]5[\/latex] minutes. A passenger boards at the bottom of the wheel. After [latex]1[\/latex] minute, what is the angle of rotation in:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">degrees?<\/li>\n<li class=\"whitespace-pre-wrap break-words\">radians?<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q68351\">Show Answer<\/button><\/p>\n<div id=\"q68351\" class=\"hidden-answer\" style=\"display: none\">\n<p>Angle of rotation after [latex]1[\/latex] minute:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Full rotation takes [latex]5[\/latex] minutes<\/li>\n<li class=\"whitespace-normal break-words\">In [latex]1[\/latex] minute, it rotates [latex]\\frac{1}{5}[\/latex] of a full circle<\/li>\n<\/ul>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>In degrees: [latex](\\frac{1}{5}) \u00d7 360\u00b0 = 72\u00b0[\/latex]<\/li>\n<li>In radians: [latex](\\frac{1}{5}) \u00d7 2\u03c0 \\text{ rad} = \\frac{2\u03c0}{5} rad \u2248 1.26 \\text{ rad}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm284090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=284090&theme=lumen&iframe_resize_id=ohm284090&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"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":"learn_it","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\/1035"}],"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":18,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1035\/revisions"}],"predecessor-version":[{"id":4469,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1035\/revisions\/4469"}],"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\/1035\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1035"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1035"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1035"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1035"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}