{“appState”:{“pageLoadApiCallsStatus”:true},”articleState”:{“article”:{“headers”:{“creationTime”:”2016-03-26T15:10:41+00:00″,”modifiedTime”:”2016-03-26T15:10:41+00:00″,”timestamp”:”2022-09-14T18:05:17+00:00″},”data”:{“breadcrumbs”:[{“name”:”Academics & The Arts”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33662″},”slug”:”academics-the-arts”,”categoryId”:33662},{“name”:”Math”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33720″},”slug”:”math”,”categoryId”:33720},{“name”:”Pre-Calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33727″},”slug”:”pre-calculus”,”categoryId”:33727}],”title”:”How to Find the Sine of a Doubled Angle”,”strippedTitle”:”how to find the sine of a doubled angle”,”slug”:”how-to-find-the-sine-of-a-doubled-angle”,”canonicalUrl”:””,”seo”:{“metaDescription”:”You use a double-angle formula to find the trig value of twice an angle. Sometimes you know the original angle; sometimes you don’t. Working with double-angle f”,”noIndex”:0,”noFollow”:0},”content”:”<p>You use a <i>double-angle formula</i> to find the trig value of twice an angle. Sometimes you know the original angle; sometimes you don’t. Working with double-angle formulas comes in handy when you’re given the sine of an angle and need to find the exact trig value of twice that angle without knowing the measure of the original angle. </p>\n<p><b><i>Note: </i></b>If you know the original angle in question, finding the sine of twice that angle is easy; you can look it up on the unit circle (shown in the figure) or use your calculator to find the answer. </p>\n<div class=\”imageBlock\” style=\”width:400px;\”><img src=\”https://www.dummies.com/wp-content/uploads/370267.image0.jpg\” width=\”400\” height=\”361\” alt=\”The whole unit circle\”/><div class=\”imageCaption\”>The whole unit circle</div></div>\n<p>However, if you don’t have the measure of the original angle and you must find the exact value of twice that angle, the process isn’t as simple. Read on!</p>\n<p>To fully understand and be able to stow away the double-angle formula for sine, you should first understand where it comes from. (The double-angle formulas for sine, cosine, and tangent are extremely different from one another, although they can all be derived by using the sum formulas.)</p>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>To find sin 2<i>x, </i>you must realize that it’s the same as sin(<i>x</i> + <i>x</i>).</p>\n </li>\n <li><p class=\”first-para\”>Use the sum formula for sine,</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370268.image1.png\” width=\”253\” height=\”49\” alt=\”image1.png\”/>\n </li>\n <li><p class=\”first-para\”>Simplify to get </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370269.image2.png\” width=\”143\” height=\”16\” alt=\”image2.png\”/>\n<p class=\”child-para\”>This formula is called the <i>double-angle formula </i>for sine. If you’re given an equation with more than one trig function and asked to solve for the angle, your best bet is to express the equation in terms of one trig function only. You often can achieve this by using the double-angle formula.</p>\n </li>\n</ol>\n<p>To solve </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370270.image3.png\” width=\”123\” height=\”19\” alt=\”image3.png\”/>\n<p>notice that it doesn’t equal 0, so you can’t factor it. Even if you subtract 1 from both sides to get 0, it still can’t be factored. So there’s no solution, right? Not quite. You have to check the identities first. The double-angle formula, for instance, says that </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370271.image4.png\” width=\”137\” height=\”16\” alt=\”image4.png\”/>\n<p>You can rewrite some things here:</p>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>List the given information.</p>\n<p class=\”child-para\”>You have </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370272.image5.png\” width=\”125\” height=\”16\” alt=\”image5.png\”/>\n </li>\n <li><p class=\”first-para\”>Rewrite the equation to find a possible identity.</p>\n<p class=\”child-para\”>You go with </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370273.image6.png\” width=\”152\” height=\”27\” alt=\”image6.png\”/>\n </li>\n <li><p class=\”first-para\”>Apply the correct formula.</p>\n<p class=\”child-para\”>The double-angle formula for sine gives you </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370274.image7.png\” width=\”124\” height=\”29\” alt=\”image7.png\”/>\n </li>\n <li><p class=\”first-para\”>Simplify the equation and isolate the trig function.</p>\n<p class=\”child-para\”>Break it down to </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370275.image8.png\” width=\”80\” height=\”19\” alt=\”image8.png\”/>\n<p class=\”child-para\”>which becomes sin 4<i>x</i> = 1/2.</p>\n </li>\n <li><p class=\”first-para\”>Find all the solutions for the trig equation.</p>\n<p class=\”child-para\”>This step gives you</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370276.image9.png\” width=\”208\” height=\”37\” alt=\”image9.png\”/>\n<p class=\”child-para\”>where <i>k</i> is an integer. Note that there are two sets of solutions because sin (pi/6) and sin (5pi/6) both equal ½. You use the notation </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370277.image10.png\” width=\”39\” height=\”17\” alt=\”image10.png\”/>\n<p class=\”child-para\”>to represent the fact that the sine function has a period of 2pi, meaning it repeats itself every 2pi units. Then you can divide everything </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370278.image11.png\” width=\”124\” height=\”20\” alt=\”image11.png\”/>\n<p class=\”child-para\”>by 4, which gives you the solutions:</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370279.image12.png\” width=\”85\” height=\”80\” alt=\”image12.png\”/>\n </li>\n</ol>\n<p>These solutions are the general ones, but sometime you may have to use this information to get to a solution on an interval.</p>\n<p>Finding the solutions on an interval is a curveball thrown at you in pre-calculus. For this problem, you can find a total of eight angles on the interval [0, 2pi). Because a coefficient was in front of the variable, you’re left with, in this case, four times as many solutions, and you must state them all. You have to find the common denominator to add the fractions. </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370280.image13.png\” width=\”247\” height=\”212\” alt=\”image13.png\”/>\n<p>Doing this one more time gets you </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370281.image14.png\” width=\”35\” height=\”37\” alt=\”image14.png\”/>\n<p>which is not in the interval [0, 2pi). Meanwhile,</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370282.image15.png\” width=\”148\” height=\”168\” alt=\”image15.png\”/>\n<p>You stop there, because adding pi/2 once more would get you a solution that is not in the interval [0, 2pi). </p>”,”description”:”<p>You use a <i>double-angle formula</i> to find the trig value of twice an angle. Sometimes you know the original angle; sometimes you don’t. Working with double-angle formulas comes in handy when you’re given the sine of an angle and need to find the exact trig value of twice that angle without knowing the measure of the original angle. </p>\n<p><b><i>Note: </i></b>If you know the original angle in question, finding the sine of twice that angle is easy; you can look it up on the unit circle (shown in the figure) or use your calculator to find the answer. </p>\n<div class=\”imageBlock\” style=\”width:400px;\”><img src=\”https://www.dummies.com/wp-content/uploads/370267.image0.jpg\” width=\”400\” height=\”361\” alt=\”The whole unit circle\”/><div class=\”imageCaption\”>The whole unit circle</div></div>\n<p>However, if you don’t have the measure of the original angle and you must find the exact value of twice that angle, the process isn’t as simple. Read on!</p>\n<p>To fully understand and be able to stow away the double-angle formula for sine, you should first understand where it comes from. (The double-angle formulas for sine, cosine, and tangent are extremely different from one another, although they can all be derived by using the sum formulas.)</p>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>To find sin 2<i>x, </i>you must realize that it’s the same as sin(<i>x</i> + <i>x</i>).</p>\n </li>\n <li><p class=\”first-para\”>Use the sum formula for sine,</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370268.image1.png\” width=\”253\” height=\”49\” alt=\”image1.png\”/>\n </li>\n <li><p class=\”first-para\”>Simplify to get </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370269.image2.png\” width=\”143\” height=\”16\” alt=\”image2.png\”/>\n<p class=\”child-para\”>This formula is called the <i>double-angle formula </i>for sine. If you’re given an equation with more than one trig function and asked to solve for the angle, your best bet is to express the equation in terms of one trig function only. You often can achieve this by using the double-angle formula.</p>\n </li>\n</ol>\n<p>To solve </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370270.image3.png\” width=\”123\” height=\”19\” alt=\”image3.png\”/>\n<p>notice that it doesn’t equal 0, so you can’t factor it. Even if you subtract 1 from both sides to get 0, it still can’t be factored. So there’s no solution, right? Not quite. You have to check the identities first. The double-angle formula, for instance, says that </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370271.image4.png\” width=\”137\” height=\”16\” alt=\”image4.png\”/>\n<p>You can rewrite some things here:</p>\n<ol class=\”level-one\”>\n <li><p class=\”first-para\”>List the given information.</p>\n<p class=\”child-para\”>You have </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370272.image5.png\” width=\”125\” height=\”16\” alt=\”image5.png\”/>\n </li>\n <li><p class=\”first-para\”>Rewrite the equation to find a possible identity.</p>\n<p class=\”child-para\”>You go with </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370273.image6.png\” width=\”152\” height=\”27\” alt=\”image6.png\”/>\n </li>\n <li><p class=\”first-para\”>Apply the correct formula.</p>\n<p class=\”child-para\”>The double-angle formula for sine gives you </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370274.image7.png\” width=\”124\” height=\”29\” alt=\”image7.png\”/>\n </li>\n <li><p class=\”first-para\”>Simplify the equation and isolate the trig function.</p>\n<p class=\”child-para\”>Break it down to </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370275.image8.png\” width=\”80\” height=\”19\” alt=\”image8.png\”/>\n<p class=\”child-para\”>which becomes sin 4<i>x</i> = 1/2.</p>\n </li>\n <li><p class=\”first-para\”>Find all the solutions for the trig equation.</p>\n<p class=\”child-para\”>This step gives you</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370276.image9.png\” width=\”208\” height=\”37\” alt=\”image9.png\”/>\n<p class=\”child-para\”>where <i>k</i> is an integer. Note that there are two sets of solutions because sin (pi/6) and sin (5pi/6) both equal ½. You use the notation </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370277.image10.png\” width=\”39\” height=\”17\” alt=\”image10.png\”/>\n<p class=\”child-para\”>to represent the fact that the sine function has a period of 2pi, meaning it repeats itself every 2pi units. Then you can divide everything </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370278.image11.png\” width=\”124\” height=\”20\” alt=\”image11.png\”/>\n<p class=\”child-para\”>by 4, which gives you the solutions:</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370279.image12.png\” width=\”85\” height=\”80\” alt=\”image12.png\”/>\n </li>\n</ol>\n<p>These solutions are the general ones, but sometime you may have to use this information to get to a solution on an interval.</p>\n<p>Finding the solutions on an interval is a curveball thrown at you in pre-calculus. For this problem, you can find a total of eight angles on the interval [0, 2pi). Because a coefficient was in front of the variable, you’re left with, in this case, four times as many solutions, and you must state them all. You have to find the common denominator to add the fractions. </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370280.image13.png\” width=\”247\” height=\”212\” alt=\”image13.png\”/>\n<p>Doing this one more time gets you </p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370281.image14.png\” width=\”35\” height=\”37\” alt=\”image14.png\”/>\n<p>which is not in the interval [0, 2pi). Meanwhile,</p>\n<img src=\”https://www.dummies.com/wp-content/uploads/370282.image15.png\” width=\”148\” height=\”168\” alt=\”image15.png\”/>\n<p>You stop there, because adding pi/2 once more would get you a solution that is not in the interval [0, 2pi). </p>”,”blurb”:””,”authors”:[{“authorId”:9703,”name”:”Yang Kuang”,”slug”:”yang-kuang”,”description”:””,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9703″}},{“authorId”:9704,”name”:”Elleyne Kase”,”slug”:”elleyne-kase”,”description”:””,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9704″}}],”primaryCategoryTaxonomy”:{“categoryId”:33727,”title”:”Pre-Calculus”,”slug”:”pre-calculus”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33727″}},”secondaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”tertiaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”trendingArticles”:null,”inThisArticle”:[],”relatedArticles”:{“fromBook”:[],”fromCategory”:[{“articleId”:262884,”title”:”10 Pre-Calculus Missteps to Avoid”,”slug”:”10-pre-calculus-missteps-to-avoid”,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/262884″}},{“articleId”:262851,”title”:”Pre-Calculus Review of Real Numbers”,”slug”:”pre-calculus-review-of-real-numbers”,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/262851″}},{“articleId”:262837,”title”:”Fundamentals of Pre-Calculus”,”slug”:”fundamentals-of-pre-calculus”,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/262837″}},{“articleId”:262652,”title”:”Complex Numbers and Polar Coordinates”,”slug”:”complex-numbers-and-polar-coordinates”,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/262652″}},{“articleId”:260218,”title”:”Special Function Types and Their Graphs”,”slug”:”special-function-types-and-their-graphs”,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/260218″}}]},”hasRelatedBookFromSearch”:true,”relatedBook”:{“bookId”:282497,”slug”:”pre-calculus-workbook-for-dummies-3rd-edition”,”isbn”:”9781119508809″,”categoryList”:[“academics-the-arts”,”math”,”pre-calculus”],”amazon”:{“default”:”https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”ca”:”https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”indigo_ca”:”http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484″,”gb”:”https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”de”:”https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20″},”image”:{“src”:”https://www.dummies.com/covers/9781119508809.jpg”,”width”:250,”height”:350},”title”:”Pre-Calculus Workbook For Dummies”,”testBankPinActivationLink”:”https://testbanks.wiley.com”,”bookOutOfPrint”:false,”authorsInfo”:”\n <p><p><b><b data-author-id=\”8985\”>Mary Jane Sterling</b></b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master’s degree in math education.</p></p>”,”authors”:[{“authorId”:8985,”name”:”Mary Jane Sterling”,”slug”:”mary-jane-sterling”,”description”:” <p><b>Mary Jane Sterling</b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master’s degree in math education.</p>”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/8985″}}],”_links”:{“self”:”https://dummies-api.dummies.com/v2/books/282497″}},”collections”:[],”articleAds”:{“footerAd”:”<div class=\”du-ad-region row\” id=\”article_page_adhesion_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_adhesion_ad\” data-refreshed=\”false\” \r\n data-target = \”[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\” id=\”du-slot-632217dd55a49\”></div></div>”,”rightAd”:”<div class=\”du-ad-region row\” id=\”article_page_right_ad\”><div class=\”du-ad-unit col-md-12\” data-slot-id=\”article_page_right_ad\” data-refreshed=\”false\” \r\n data-target = 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