[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572137-105":53,"doc-detail-572137-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","trigonometry-and-modelling-chapter-7-cheat-sheet","Trigonometry and modelling - Chapter 7 - Cheat Sheet","","Trigonometry and modelling chapter cheat sheet focused on the key addition, subtraction, and double-angle identities for sin, cos, and tan. It explains how to use the formulae to compute exact trigonometric values for special angles, and how to prove double-angle results using angle addition identities and coefficient comparison. Worked examples also show simplification into single trigonometric terms, solving expressions via simplification and CAST/graphs, and proving identities by transforming one side of an equation into the other.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/trigonometry-and-modelling-chapter-7-cheat-sheet/572137/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/trigonometry-and-modelling-chapter-7-cheat-sheet/572137.png","ImageObject",442,249,{"name":88,"@type":89},"DoDO","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you find exact values like sin(75°) using the cheat sheet?","Question",{"text":108,"@type":109},"Rewrite the target angle as a sum of angles with known exact values (e.g., 75° = 45° + 30°). Substitute sin(45°), cos(45°), sin(30°), and cos(30°) into the relevant addition formula, then simplify.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What approach is used to prove the double-angle sine and cosine formulae?",{"text":113,"@type":109},"Use the addition formulae to express sin(2θ) or cos(2θ) as sin(θ+θ) and cos(θ+θ). Expand, then equate coefficients (or use identities such as sin²θ+cos²θ=1) to derive the standard double-angle forms.",{"name":115,"@type":106,"acceptedAnswer":116},"How can complicated trigonometric equations be solved effectively?",{"text":117,"@type":109},"First simplify the equation using the provided formulae, then solve the resulting simpler equation. The sheet illustrates using exact values, dividing by common factors, finding principal solutions, and using CAST or graphical methods to list all solutions within a given interval.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572137,1792052364,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},962093270745,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Trigonometry and modelling Cheat Sheet  \nThis chapter builds upon the previous, introducing more useful methods, formulae and identities relating to trigonometric functions  \n▪ sin (􀜣 + 􀜤) ≡ 􀝏􀝅􀝊􀜣􀜿􀝋􀝏􀜤 + 􀜿􀝋􀝏􀜣􀝏􀝅􀝊􀜤  \n▪ cos (􀜣 + 􀜤) ≡ 􀜿􀝋􀝏􀜣􀜿􀝋􀝏􀜤 − 􀝏􀝅􀝊􀜣􀝏􀝅􀝊􀜤  \n▪ tan(􀜣 + 􀜤) ≡ 􀝐1􀜣􀝐􀜽􀝐􀜣􀜽􀝊􀝐􀜽􀜤􀝊􀜤  \n▪ 􀝏􀝅􀝊 (􀜣 − 􀜤) ≡ 􀝏􀝅􀝊􀜣􀜿􀝋􀝏􀜤 − 􀜿􀝋􀝏􀜣􀝏􀝅􀝊􀜤  \n▪ 􀜿􀝋􀝏 (􀜣 − 􀜤) ≡ 􀜿􀝋􀝏􀜣􀜿􀝋􀝏􀜤 + 􀝏􀝅􀝊􀜣􀝏􀝅􀝊􀜤  \n▪ 􀝐􀜽􀝊(􀜣 − 􀜤) ≡ 􀝐1􀜣􀝐􀜽􀝐􀜣􀜽􀝊􀝐􀜽􀜤􀝊􀜤  \nYou need to know how to use the above formulae to find exact values of trigonometric functions for various angles.  \nExample 1: Show, using the formula for sin( 􀜣 + 􀜤), that sin( 75°) = ~~√~~6+4~~√~~2  \nWe can rewrite sin( 75°) as sin( 45° + 30°) . We choose 45 and 30 because we know the exact values of sin( 45°) , cos( 45°) , sin( 30°) 􀜽􀝊􀝀 cos(30°), so when we put them into the addition formula, we will have all terms given as exact values.  \nsin( 75°) ≡ sin( 45 + 30)  \nsin( 45 + 30) ≡ 􀝏􀝅􀝊45􀜿􀝋􀝏30 + 􀜿􀝋􀝏45􀝏􀝅􀝊30  \n= ቆ2ቇ ቆ3ቇ + ቆ2ቇ ൬12൰ = 6 + 2  \n√6 + √2  \n=  \n4  \n▪ 􀝏􀝅􀝊(2􀜣) ≡ 2􀝏􀝅􀝊􀜣􀜿􀝋􀝏􀜣  \n▪ cos(2􀜣) ≡ 􀜿􀝋􀝏 2􀜣 − 􀝏􀝅􀝊2􀜣 = 1 − 2􀝏􀝅􀝊2􀜣 = 2􀜿􀝋􀝏 2􀜣 − 1  \n▪ tan(2􀜣) ≡ 1~~ ~~􀝐􀜽􀝐􀝊􀜣􀜽􀝊2􀜣  \nYou can be asked to reproduce these proofs!  \n\n| Example 2: Using the addition formulae, prove each of the above double-angle formulae. |  |\n| --- | --- |\n| Proving the double-angle sine formula:\u003Cbr>Proving the double-angle cosine formula:\u003Cbr>Using 􀝏􀝅􀝊2􀜣 + 􀜿􀝋􀝏 2􀜣 ≡ 1 to prove the other cosine double angle formulae:\u003Cbr>Proving the double-angle tangent formula: | 􀝏􀝅􀝊 ( 2􀜣) = 􀝏􀝅􀝊 ( 􀜣 + 􀜣) = 􀝏􀝅􀝊􀜣􀜿􀝋􀝏􀜣 + 􀜿􀝋􀝏􀜣􀝏􀝅􀝊􀜣= 2􀝏􀝅􀝊􀜣􀜿􀝋􀝏􀜣\u003Cbr>cos( 2􀜣) = 􀜿􀝋􀝏 ( 􀜣 + 􀜣) = 􀜿􀝋􀝏􀜣􀜿􀝋􀝏􀜣 − 􀝏􀝅􀝊􀜣􀝏􀝅􀝊􀜣\u003Cbr>= 􀜿􀝋􀝏 2􀜣 − 􀝏􀝅􀝊2􀜣 By replacing 􀜿􀝋􀝏 2􀜣 with 1 − 􀝏􀝅􀝊2􀜣:\u003Cbr>⇒ cos( 2􀜣) = 1 − 2􀝏􀝅􀝊2􀜣 Also, by replacing 􀝏􀝅􀝊2􀜣 with 1 − 􀜿􀝋􀝏 2􀜣:\u003Cbr>⇒ cos( 2􀜣) = 2􀜿􀝋􀝏2􀜣 − 1  􀝐􀜽􀝊􀜣 + 􀝐􀜽􀝊􀜣 \u003Cbr>tan( 2􀜣) = tan( 􀜣 + 􀜣) =\u003Cbr>1 − 􀝐􀜽􀝊􀜣􀝐􀜽􀝊􀜣\u003Cbr>2􀝐􀜽􀝊􀜣= ~~ ~~\u003Cbr>1 − 􀝐􀜽􀝊2􀜣 |\n\nYou can see that there are three different versions for the cosine double angle formula. It is important you are familiar with all three as one may be more useful than the others in certain questions.  \nExample 3: Simplify as much as possible the expression: 􀝏􀝅􀝊4􀝔 − 2􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 + 􀜿􀝋􀝏 4􀝔  \nSpotting the factorisation: 􀝏􀝅􀝊4􀝔 − 2􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 + 􀜿􀝋􀝏 4􀝔 =  \n(􀜿􀝋􀝏 2􀝔 − 􀝏􀝅􀝊2􀝔)2  \nUsing 􀜿􀝋􀝏2􀝔 = 􀜿􀝋􀝏 2􀝔 − 􀝏􀝅􀝊2􀝔: = (􀜿􀝋􀝏2􀝔)2 = 􀜿􀝋􀝏 22􀝔  \nExample 4: Simplify as much as possible the expression: √1 + 􀜿􀝋􀝏􀝔  \nSince 􀜿􀝋􀝏2􀝔 = 2􀜿􀝋􀝏 2􀝔 − 1  \nSubstituting this result into the given expression:  \n􀜿􀝋􀝏􀝔 = 2􀜿􀝋􀝏 2 (􀝔2) − 1   ⇒ √1 + 􀜿􀝋􀝏􀝔 = ට1 + 2􀜿􀝋􀝏2 (􀝔2) − 1 = ට2􀜿􀝋􀝏2 (􀝔2)  \nExpressions of the above form can be simplified into one trigonometric term.  \n▪ 􀜽􀝏􀝅􀝊􀝔 ± 􀜾􀜿􀝋􀝏􀝔 can be expressed as 􀜴􀝏􀝅􀝊(􀝔 ± 􀟙)  \n▪ 􀜽􀜿􀝋􀝏􀝔 ± 􀜾􀝏􀝅􀝊􀝔 can be expressed as 􀜴􀜿􀝋􀝏(􀝔 ∓ 􀟙) where 􀜽, 􀜾 , 􀜴 > 0 􀜽􀝊􀝀 0 \u003C 􀟙 \u003C 􀰗2 .  \nThe procedure for achieving the above simplifications can be broken down into three steps:  \n[1] Expand the form using the addition formulae, and equate it to 􀜽􀝏􀝅􀝊􀝔 ± 􀜾􀜿􀝋􀝏􀝔  \n[2] Compare the coefficients of 􀝏􀝅􀝊􀝔 and 􀜿􀝋􀝏􀝔 on both sides of the equation, to get two equations in terms of 􀜴 and 􀟙 .  \n[3] Solve these simultaneously to find 􀜴 and 􀟙 .  \n| Example 5: Express 􀜿􀝋􀝏2􀝔 − 2􀝏􀝅􀝊2􀝔 in the form 􀜴􀜿􀝋􀝏(2􀝔 + 􀟙), where 􀜴 > 0 􀜽􀝊􀝀 0 \u003C 􀟙 \u003C 􀰗2 |  |\n| --- | --- |\n| Proving the double-angle sine formula: Equating coefficients:\u003Cbr>Solving simultaneously.:\u003Cbr>We divide equation [2] by [1].\u003Cbr>Finding 􀜴:\u003Cbr>Square equations [1] and [2] then add them together. We also use the identity 􀜿􀝋􀝏 2􀟙 + 􀝏􀝅􀝊2 􀟙 ≡ 1\u003Cbr>Putting everything together: | 1􀜿􀝋􀝏2􀝔 − 2􀝏􀝅􀝊2􀝔 = 􀜴􀜿􀝋􀝏 ( 2􀝔 + 􀟙) ≡ 􀜴􀜿􀝋􀝏2􀝔 􀜿􀝋􀝏􀟙 − 􀜴􀝏􀝅􀝊2􀝔 􀝏􀝅􀝊􀟙\u003Cbr>1 = 􀜴􀜿􀝋􀝏􀟙 (1) (equating 􀜿􀝋􀝏2􀝔 coefficients)\u003Cbr> − 2 = 􀜴􀝏􀝅􀝊􀟙 (2) (equating 􀝏􀝅􀝊2􀝔 coefficients)􀜴􀝏􀝅􀝊􀟙   − 2\u003Cbr>􀝐􀜽􀝊􀟙 = = = 2 􀜴􀜿􀝋􀝏􀟙 1\u003Cbr>∴ 􀟙 = arctan(2) = 1.11 (1)2 + (2)2 ⇒ 􀜴 2􀜿􀝋􀝏 2􀟙 + 􀜴 2􀝏􀝅􀝊2􀟙 = (1)2 + 2)2 ⇒ 􀜴 2( 􀜿􀝋􀝏2􀟙 + 􀝏􀝅􀝊2􀟙) = 5\u003Cbr>⇒ 􀜴 2 = 5 ∴ 􀜴 = √5\u003Cbr>\u003Cbr>So 􀜿􀝋􀝏2􀝔 − 2􀝏􀝅􀝊2􀝔 = √5cos(2􀝔 + 1. 11) |\n\nThis form is often useful because it makes solving equations and finding minimum/maximum values much easier.  \nExample 6: Given that 􀝃( 􀝔) =  18   \n50 + 􀜿􀝋􀝏2􀝔 − 2􀝏􀝅􀝊2􀝔,  \ncalculate: (i) the maximum value of 􀝃 ( 􀝔) .  \n(ii) The smallest positive value of x at ","cbCaiuiL4hNhncOW","https://ap.wps.com/l/cbCaiuiL4hNhncOW","pdf",1640856,"English","# Trigonometry and modelling cheat sheet\n## Core addition/subtraction identities\n## Exact values for special angles\n## Double-angle formula proofs\n## Simplifying to single trigonometric terms\n## Solving trigonometric equations\n## Proving identities","[{\"question\":\"How do you find exact values like sin(75°) using the cheat sheet?\",\"answer\":\"Rewrite the target angle as a sum of angles with known exact values (e.g., 75° = 45° + 30°). Substitute sin(45°), cos(45°), sin(30°), and cos(30°) into the relevant addition formula, then simplify.\"},{\"question\":\"What approach is used to prove the double-angle sine and cosine formulae?\",\"answer\":\"Use the addition formulae to express sin(2θ) or cos(2θ) as sin(θ+θ) and cos(θ+θ). Expand, then equate coefficients (or use identities such as sin²θ+cos²θ=1) to derive the standard double-angle forms.\"},{\"question\":\"How can complicated trigonometric equations be solved effectively?\",\"answer\":\"First simplify the equation using the provided formulae, then solve the resulting simpler equation. The sheet illustrates using exact values, dividing by common factors, finding principal solutions, and using CAST or graphical methods to list all solutions within a given interval.\"}]","Trigonometry and modelling - Chapter 7 - Cheat Sheet | PDF",1791925746]