{"id":5707,"date":"2022-05-06T20:55:30","date_gmt":"2022-05-07T03:55:30","guid":{"rendered":"https:\/\/studysoup.com\/blog\/?p=5707"},"modified":"2022-05-06T20:55:30","modified_gmt":"2022-05-07T03:55:30","slug":"solved-trigonometric-identities-problem","status":"publish","type":"post","link":"https:\/\/newblog.studysoup.com\/?p=5707","title":{"rendered":"Solved: Trigonometric Identities Problem"},"content":{"rendered":"\n<p>If you\u2019re taking a geometry or trigonometry class, you\u2019ll most likely learn trigonometric identities.&nbsp;<br><\/p>\n\n\n\n<p>In simple terms, trigonometric identities are equations that are related to different trigonometric functions and are true for any value of the variable. It could be hard since there are numerous trigonometry identities.<br><\/p>\n\n\n\n<p>In this post you\u2019re going to how to solve one of the top asked examples of trigonometric identities. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Solved: Trigonometric Identities Problem<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-embed-youtube aligncenter wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Find a Trigonometric Expression Value Given The Sine Sum Identity Expression\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q_7JW63vVVs?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>                                Find the exact value of:<\/strong><\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/JDwkImGxDrNXfFmUd4lXdoRrmdfQgIhkGCj7I2J78owXYzGmlqUlaoG8vJyUp2XyvKNIdvg18mLdNhZMfLeUtpFYB9389Kz0uhcUlZcfxWqO-z7RhwIMf4aDYZJkwTbhER78U7ntCdkDIqQ3bQ\" alt=\"\" width=\"355\" height=\"32\"\/><\/figure><\/div>\n\n\n\n<p>In this problem we\u2019re asked to find the exact value of the given expression. Right away, the given expression should remind us of the sum and difference identities for sine and cosine shown below:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/HE_52XCCIxcwyj5oEv4s4sfI72KRAhAiadr1DW2LzVZzNBrtOrUdNMS58CKExPlQv3plr5TeIP2vwofP72nlLpOlMzaIwE6QKxtEknD5HTBIPEGhOt5sbwCu35aTDOf6cWtDA6FVTKMxW0OdqA\" alt=\"\" width=\"519\" height=\"27\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/emnFFyN_pjxT5FjNndcQIJLuO6ZK_ECbaMo1Hj_P4AQet4XS54B4VhON3qpfJsjlRW70GM0XjrvY6sVd30Fo00nq4EPoGAGhiodNDAQs1Ffv8UipgJoMkpqm7Cz5mYO0EyIcW7ujO5PzimZ7WA\" alt=\"\" width=\"527\" height=\"27\"\/><\/figure><\/div>\n\n\n\n<p>For our expression, notice A is equal to pi divided by 3 and B is equal to pi divided by 2. Due to the fact we have a sum, the expression is equal to the sine of the sum of the angles.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/XloIDNuIh1oKe5CCO6p6n7rlljV6S74NDzajw4C1RaU5Mvm9-Wqim_pdjPHGqtvyjLKvlw1Vtx3xtPWul0FzKbLn8MaxQMyaFDHbvjo8VdWOnjK2drltE0lWoxxkwePMPv7pcrYQ7b0VEXr-Ww\" alt=\"\" width=\"420\" height=\"38\"\/><\/figure><\/div>\n\n\n\n<p>1) First, we have to find the sum of the angles and then determine the sine function value.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/QONX25lKwMyGwkk_44IXW6-GdMbSpJbWr9QGCA2kkJLh_mh23RGY3NtjUMCP0L6hn82yF9I9j_4N4-BzQJMuXmJxbYaDdzdkV0kLG8w2PLLGa8TlVayKTr-ZAizqk5x1AQv-s8pYeamQSJ1vQQ\" alt=\"\" width=\"503\" height=\"30\"\/><\/figure><\/div>\n\n\n\n<p>To add the fractions we have to find a common denominator. In this case, the least common denominator is a least common multiple of 2 and 3, which is 6. Therefore, we multiply the numerator and denominator of pi over 3 by 2, and we multiply the numerator and denominator of pi over 2 by 3:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/RCDg1malS0_TgLwBwfh7JNPbjJ_d08qdicw7X20gbVySBgklq6Kn1BDkSww7HhreeMNP6OnnYTVQadyEPg11ufMr3wLJaQgp4zxG2rF51y3z8QDOHrLNe12Zl1Zzu2DT1F5WdcPBBBcPdz6fsw\" alt=\"\" width=\"256\" height=\"47\"\/><\/figure><\/div>\n\n\n\n<p>2) Now we know:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/ar6I_uKVMNKr6ncBrI9ucpbze1daR0wpfQVvZOxVecSfHQ2uGohz-Og9fCai9g3YORAeXRjnFvX59e5tMq2_PlYJdwKSaaT6W4hhgXaqIt0pUqxoDP6FRAXO7ne8BEAOG9-oF6GbJQ7N7vBlVA\" alt=\"\" width=\"299\" height=\"40\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><a href=\"https:\/\/bit.ly\/3vNr1Nk\" target=\"_blank\" rel=\"noreferrer noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/studysoup.com\/blog\/wp-content\/uploads\/2022\/04\/StudySoup-Textbook-Survival-Guides.png\" alt=\"textbooksurvivalguides\" class=\"wp-image-5657\" width=\"741\" height=\"247\"\/><\/a><\/figure><\/div>\n\n\n\n<p>We need to find the sine function value, which we can do using a reference triangle, or the unit circle. To use a reference triangle, we first sketch 5\/6 pi radians in standard position.&nbsp;<br><\/p>\n\n\n\n<p>The initial side is along the positive x-axis, and then we rotate 5 pi divided by 6 radians counterclockwise, which is 1\/6 pi, or pi divided by 6 radians short of 1\/2 a rotation, which would be approximately here. this angle here is 5 pi divided by 6 radians, and the reference angle is the angle formed with the terminal side and this negative x-axis, which is pi over 6 radians:<br><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/0iR36947eInNjdSARh7fGySW659yrXwpvlbX5bcwVEYDvQOEPapns3bd_E19J18O8C1-YQqlGhuNwF55AI7KsSCYu2F-bMfHIXv87JXKzcbw9wS4pZ3Ine0RyEUwpDCXbf-2V_fztsCao3Vccg\" alt=\"\" width=\"511\" height=\"458\"\/><\/figure><\/div>\n\n\n\n<p>3) Let\u2019s sketch the reference triangle. We should recognize this as a 30-60-90 reference triangle, where we can label the short leg 1, the hypotenuse 2, and the longer leg square root of 3.<br><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/IV6tj2LIQKzqK_e-zYcHQyw0S26JICxxGwbWoeH2ccgWyR3yPgnF7oXg_afvGgmQTzudMCfC_pmktbBwuKZZcFgJgudn0A1cs9dtzURmWEDHlHHcU_9LVpji9m7NmjHlpyqbSiVYKND1Udn6Rg\" alt=\"\" width=\"487\" height=\"455\"\/><\/figure><\/div>\n\n\n\n<p>However, we are in the second quadrant where x is negative, and therefore, the square root of 3 is negative. We can use this reference triangle to determine the sine function value, where the sine of 5 pi divided by 6 radians is equal to the ratio of the opposite side to the hypotenuse, which we can see is equal to 1 divided by 2, or 1\/2. So, the exact value of the given expression is equal to 1\/2.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/_cM_n2N8BnIqZ4R9K9xVLTY5UmezAY8VubUcBg94-ADwBhmbR0eniGF5QKMqQmJb1MAPTqsllrARQi1civ1kUH_hQREBQ5TWcA9eySDJBpbf4_TqB9mf-AMYq0Yy1GymiReif_SCjSWI_TzSWw\" alt=\"\" width=\"191\" height=\"47\"\/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/O4-OD6ptpUpCdws7xTvXP8XUlZ-Dr38iYud2ZZq7olCvxGmRz9Buh0VNMaLGWB-X7uK2vgR8vnpEU5j53p-K5lQW647Tqh8YvmR6mv5A5ywQEajmSCAhZMLCularONuMG5wS_PWGt_MYB4gSxQ\" alt=\"\" width=\"510\" height=\"456\"\/><\/figure><\/div>\n\n\n\n<p>4) Let\u2019s also find this sine function value on the unit circle. The terminal side of 5 pi divided by 6 radians intersects the unit circle. On the unit circle, remember sine theta is equal to y, and notice how the y-coordinate is equal to 1\/2.&nbsp;<br><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/PyixjnfIwDnWu_p0-KdMslzWWNQwE3bC2GZWZVMX6e7ZyrPZ6dcdrA6CZH9Xr7NMX7yOVfBo1UpZIckQeSY8e3g7-YVCV9fb5LB0BweSx_QBUXmLfuewm050DN-4lozmpeNsIcH-mIlxyh2PFQ\" alt=\"\" width=\"471\" height=\"471\"\/><figcaption>source:<a href=\"https:\/\/en.wikipedia.org\/wiki\/Unit_circle#:~:text=In%20mathematics%2C%20a%20unit%20circle,system%20in%20the%20Euclidean%20plane.\">Wikipedia\ufeff<\/a><\/figcaption><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Conclusion<\/strong><\/h2>\n\n\n\n<p>Trig identities can be overwhelming- But that\u2019s where a <a href=\"https:\/\/bit.ly\/3vNr1Nk\" target=\"_blank\" rel=\"noreferrer noopener\" aria-label=\"high-quality textbook survival guide (opens in a new tab)\">high-quality textbook survival guide<\/a> comes in.With it, you\u2019ll not only be able to get the right answers, but understand concepts to prepare for the exams quickly without stress. <br><\/p>\n\n\n\n<p><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>If you\u2019re taking a geometry or trigonometry class, you\u2019ll most likely learn trigonometric identities.&nbsp; In simple terms, trigonometric identities are equations that are related to different trigonometric functions and are true for any value of the variable. It could be hard since there are numerous trigonometry identities. In this post you\u2019re going to how to [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":5716,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[10,18,19],"class_list":["post-5707","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math","tag-math","tag-trigonometric-identities","tag-trigonometry"],"_links":{"self":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts\/5707","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5707"}],"version-history":[{"count":0,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/wp\/v2\/posts\/5707\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=\/"}],"wp:attachment":[{"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5707"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5707"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/newblog.studysoup.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}