[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573394-105":53,"doc-detail-573394-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","gr-12-maths-bookwork-all-the-proofs-you-need-to-know-paper-1-paper-2-proofs","Gr 12 Maths BOOKWORK - All the proofs you need to know! - Paper 1 & Paper 2 Proofs","","“The Answer Series” Gr 12 Maths Bookwork provides proof-focused extracts aligned to the TAS Grade 12 Maths structure for 2023/2024. It covers Paper 1 proofs on arithmetic and geometric series with sum-to-n term derivations, including infinite series conditions, and Paper 2 proofs related to geometry theorems and trigonometry proofs plus formulae derivations. It also includes calculator instructions and emphasizes understanding through essential practice with graded exercises and exam questions.",{"@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/gr-12-maths-bookwork-all-the-proofs-you-need-to-know-paper-1-paper-2-proofs/573394/",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/gr-12-maths-bookwork-all-the-proofs-you-need-to-know-paper-1-paper-2-proofs/573394.png","ImageObject",442,249,{"name":88,"@type":89},"Margaret","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-14",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},"What topics are covered in Paper 1?","Question",{"text":108,"@type":109},"Paper 1 focuses on proofs for arithmetic and geometric series, including expressions for the sum of n terms and related identities.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which trigonometry formula derivations are included?",{"text":113,"@type":109},"The document derives compound angle formulae (cos and sin of sums and differences) and double angle formulae for sin 2A and cos 2A.",{"name":115,"@type":106,"acceptedAnswer":116},"Are there recommendations for practice and exam preparation?",{"text":117,"@type":109},"Yes. It states that practice is essential and directs readers to graded exercises and copious exam questions in TAS Grade 12 study guides.","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},573394,1792053257,{"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":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":76},137451207643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","The Answer Series  \nGr 12 Maths BOOKWORK  2023/2024 (Extracts from TAS Gr 12 Maths 2-in-1 and Toolkit)  \nAll the proofs you need to know!  \n(also available in Afrikaans)  \nPaper 1: (a maximum of 6 marks) page 1  \nArithmetic & Geometric Series  \nPaper 2: (a maximum of 12 marks) page 2 to 7  \nGeometry (7 theorems) and  \nTrigonometry (4 proofs & 5 formulae derivations)  \nPlus Calculator Instructions  \nPAPER 1: PROOFS  \nArithmetic and Geometric Series  \nSn : the sum of n terms  \n􀀂 Arithmetic Series:  \nSn = a + (a + d) + (a + 2d) +   + Tn   n terms  \n& Sn = Tn + (Tn – d) + (Tn – 2d) +   + a   n terms  \n∴ 2 Sn = (a + Tn) + (a + Tn) + (a + Tn) +   + (a + Tn)   n terms  \n∴ 2 Sn = n (a + Tn)∴ Sn = n2 (a + Tn) 􀀂  \nBut T n = a + (n – 1)d ∴ Sn = n2 [ a + a + (n – 1)d ]  \n∴ Sn = n2 [2a + (n – 1)d] 􀀂  \n􀀂 Geometric Series:  \nSn = a + ar + ar2 + ar3 +   + arn – 2 + arn – 1   ➊  \n% r) ∴ rS n = ar + ar2 + ar3 +   + arn – 1 + arn   ➋  \n the 'middle bit'falls away.  \n➊ – ➋: ∴ Sn – rS n = a – arn  \n∴ Sn(1 – r) = a(1 – rn )  \n∴ Sn = a(11−~~ ~~rr~~ ~~n~~ ~~)~~ ~~ . . . for r \u003C 1 or ×~~ ~~((~~ ~~−~~ ~~11)) : Sn = a(rr1) . . . for r > 1  \nAlso remember:  \n􀀂 S∞ = Sn as n 􀀃 ∞ if –1 \u003C r \u003C 1 = a~~ ~~(11~~ ~~0r)~~ ~~ . . . rn 􀀃 0 if –1 \u003C r \u003C 1 = 1~~ ~~~~ ~~r  \n􀀂 k∑ 1Tk  \n= Sn and k1Tk = S ∞ if –1 \u003C r \u003C 1  \n1 Copyright © The Answer Series  \nDerivation of Compound Angle Formulae  \nWe accept and use the formula:  \ncos(A-- B) = cos A cos B + sin A sin B  \n The cos of the difference  \n􀀂 cos(A + B) = cos [A-- (–B)]   \nof the two angles  \n= cos A cos(–B) + sin A sin(–B)  \n= cos A cos B + sin A (–sin B)  \n= cos A cos B -- sin A sin B 􀀂  \n􀀂 sin(A + B) = cos[90º – (A + B)]  \n= cos[90º – A – B]  \n The cos of the difference  \n= cos[(90º– A) -- B]   \nof the two angles  \n= cos(90º– A) cos B + sin(90º – A) sin B  \n= sin A cos B + cos A sin B 􀀂  \n􀀂 sin(A--B) = cos[90º–(A – B)]  \n= cos[90º – A + B]  \n The cos of the difference  \n= cos[(90º– A) -- (–B)]   \nof the two angles  \n= cos(90º– A) cos(–B) + sin(90º– A) sin(–B)  \n= sin A cos B + cos A(–sin B)= sin A cos B -- cos A sin B 􀀂  \nPractice is essential !  \nSee our graded exercises and copious exam questions in TAS Grade 12 study guides.  \n7  \nDerivation of Double Angle Formulae  \n􀀂  \nsin 2A  \nsin 2A = sin(A + A) . . . see sin(A + B) above  \n= sin A cos A + cos A sin A . . . add like terms: xy + y x = 2xy = 2 sin A cos A 􀀂  \n􀀂  \ncos 2A  \ncos 2A = cos(A + A) . . . see cos(A + B) above = cos A cos A – sin A sin A  \n= cos2A--sin2A 􀀂  \nBut: sin2A + cos2A = 1  \n∴ cos2A = 1 – sin2A & sin2A = 1 – cos2A  \nâ cos 2A = (1 – sin2A)– sin2A = 1--2 sin2A 􀀂  \nOR  \ncos 2A = cos2A –(1 – cos2A)= cos2A – 1 + cos2A = 2 cos2A-- 1 􀀂  \nIt is easier to memorise formulae which we understand.  \nCopyright © The Answer Series","cbCaimTvsu0hb26Y","https://ap.wps.com/l/cbCaimTvsu0hb26Y","pdf",13890177,9,"English","# Arithmetic & Geometric Series\n## Arithmetic series sum derivations\n## Geometric series sum derivations\n# Derivation of Compound Angle Formulae\n## cos(A - B) and cos(A + B)\n## sin(A + B) and sin(A - B)\n# Derivation of Double Angle Formulae\n## sin 2A\n## cos 2A\n# Practice and exam preparation","[{\"question\":\"What topics are covered in Paper 1?\",\"answer\":\"Paper 1 focuses on proofs for arithmetic and geometric series, including expressions for the sum of n terms and related identities.\"},{\"question\":\"Which trigonometry formula derivations are included?\",\"answer\":\"The document derives compound angle formulae (cos and sin of sums and differences) and double angle formulae for sin 2A and cos 2A.\"},{\"question\":\"Are there recommendations for practice and exam preparation?\",\"answer\":\"Yes. It states that practice is essential and directs readers to graded exercises and copious exam questions in TAS Grade 12 study guides.\"}]","Gr 12 Maths BOOKWORK - All the proofs you need to know! - Paper 1 & Paper 2 Proofs | PDF",1791938841]