[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571380-105":53,"doc-detail-571380-en":119},{"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":112,"head_meta":114,"extra_data":116,"updated_unix":118},105,"en","trigonometric-identities-edexcel-pure-year-1-angles-and-unit-circle","Trigonometric identities - Edexcel Pure Year 1 - Angles and unit circle","","A study-focused cheat sheet for Edexcel Pure Year 1 on trigonometric identities, covering angles across all four quadrants and the use of the unit circle to determine the signs of sin, cos, and tan. Includes guidance for angles outside 0–360° using equivalence modulo 360°, worked examples expressing trigonometric values using acute angles, and rules for transforming expressions like sin(180° ± θ) and tan shifts. Also presents exact values, identity-based simplification, and step-by-step solving methods for simple, harder, and quadratic trigonometric equations in specified intervals.",{"@graph":63,"@context":111},[64,80,102],{"@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":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/trigonometric-identities-edexcel-pure-year-1-angles-and-unit-circle/571380/",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/trigonometric-identities-edexcel-pure-year-1-angles-and-unit-circle/571380.png","ImageObject",442,249,{"name":88,"@type":89},"Dusta123","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-27","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",8,{"@type":103,"mainEntity":104},"FAQPage",[105],{"name":106,"@type":107,"acceptedAnswer":108},"How are simple trigonometric equations solved in this cheat sheet?","Question",{"text":109,"@type":110},"Solve equations in forms like sin θ = k, cos θ = k, or tan θ = k within a specified interval. The method includes rearranging to match the target function, finding principal values, and then checking which quadrants produce valid solutions.","Answer","https://schema.org",{"og:url":78,"og:type":113,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":115,"canonical":78},"index,follow",{"doc_id":117,"site_id":56},571380,1792009054,{"code":4,"msg":5,"data":120},{"doc_id":117,"user_id":121,"nickname":88,"user_avatar":122,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":123,"file_id":124,"file_url":125,"file_type":126,"file_size":127,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":128,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":129,"faqs":130,"seo_title":131,"seo_description":61,"update_tm":132,"read_time":4},962093541925,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Trigonometric identities Cheat Sheet  \nEdexcel Pure Year 1  \nAngles in all four quadrants  \nUnit circles:  \nA unit circle is a circle with radius of 1 unit. It will help you understand the trigonometric ratios.  \nFor a point 􀀭 (􀀡, 􀀢) on a unit circle such that 􀀰􀀭 making an angle with the positive 􀀡-axis cos 􀀦 = 􀀡-coordinate of 􀀭  \nsin 􀀦 = 􀀢 -coordinate of 􀀭  \ntan 􀀦 = !\" =gradient of 􀀰􀀭  \nYou always start measuring 􀀦 from positive xaxis  \nPositive angles  Anti-clock wise Negative angles  Clockwise  \nWith the help of unit circle you can find values and signs of sine, cosine and tangent.  \nAngles may lie outside the range 0-360°, but they always lie in oneof the four quadrants.  \nFor e.g. 520° is equivalent to 520° − 360° = 160° which lies in second quadrant  \nFind the signs of sin 􀀦 , cos 􀀦 and tan 􀀦 in the second quadrant.  \nDraw a circle with centre 0 and radius 1, with 􀀭(􀀡, 􀀢) in the second quadrant.  \nYou know that 􀀡 is −􀀺􀀻 and 􀀢 is +ve in the second quadrant  \nsin 􀀦 = +􀀺􀀻 , cos 􀀦 = −ve  \ntan 􀀦 = 􀀺􀀺􀀻􀀻 = −􀀺􀀻  \nSo, only sin 􀀦 is +􀀺􀀻 in the second quadrant  \nWith the help of the following diagram, you can determine the signs of each of the trigonometric ratios  \nOnly 􀀿􀁀􀁁 􀁂 is positive for angle 􀀦 in the second quadrant.  \nsin 􀀦 , cos 􀀦 and tan 􀀦 are all positive for angle 􀀦 in the first quadrant.  \nOnly 􀁃􀁄􀁁 􀁂 is positive  \nfor angle 􀀦 in the third quadrant  \nOnly 􀁅􀁆􀀿 􀁂 is positive for angle 􀀦 in the fourth quadrant.  \nYou can use the following rules to find sin, cos or tan of any positive or negative angle using the corresponding acute angle made with the 􀀡-axis  \nsin (180° − 􀀾) = + sin 􀀾  \nsin (180° + 􀀾) = − sin 􀀾  \nsin(360° − 􀀾) = − sin 􀀾  \ntan (180° − 􀀾) = − tan 􀀾  \ntan (180° + 􀀾) = + tan 􀀾 tan(360° − 􀀾) = − tan 􀀾  \nExample2:  \n180° − 􀀦  \n180°+􀀦  \ncos (180° − 􀀾) = − cos 􀀾  \ncos (180° + 􀀾) = − cos 􀀾 cos (360° − 􀀾) = cos 􀀾  \nExpress the following in terms of trigonometric ratios of acute angles.  \na. sin 240° b. cos(−50°)  \na. b.  \nThe angle 240° is obtuse and measured from the +􀀦􀀧 􀀩 -axis anti-clockwise.  \nSo the acute angle is 60° sin is – 􀀦􀀧 in the third quadrant So sin 240° = − sin 60°  \nThe angle −50° is the angle measured from the positive 􀁒 -axis clockwise.  \n50° is the acute angle.  \ncos is +􀁙􀁚 in the fourth quadrant So cos(−50°) = cos 50°  \nExample 3:  \nGiven that 􀀡 is an acute angle, express tan(􀀡 − 540°)in terms of tan 􀀡  \nTo express tan(􀀡 − 540°) in terms of tan 􀀡 , we need to find in which quadrant the angle 􀀡 − 540° lies.  \nYou know that 540° is equivalent to 540° − 360° = 180°  \n⇒ −540° is equivalent to −180° ⇒ 180° clockwise and 􀀡 = anti-clockwise  \nSo first you will go 180° clockwise and then 􀀡 tan is +ve in the third quadrant  \nHence, tan(􀀡 − 540°) = tan 􀀡  \nExact values of trigonometric ratios.  \nanti-clockwise  which will be in the third quadrant.  \nYou can find exact values of sin, cos and tan of 30°, 45° and 60° . Please refer the table below for the exact values.  \n\n|  | 􀀮􀀯 ° | 􀀰􀀱 ° | 􀀲􀀯 ° |\n| --- | --- | --- | --- |\n| 􀀳􀀴􀀵 􀀶 | 1 2 | 1 √2\u003Cbr>| √3\u003Cbr>2 |\n| 􀀷􀀸􀀳 􀀶 | √3\u003Cbr>2 | 1 √2\u003Cbr>| 1 2 |\n| 􀀹􀀺􀀵 􀀶 | 1  √3\u003Cbr>~~  ~~ =\u003Cbr>√3 3 | 1 | √3 |\n\nTrignometric Identities:  \nEquation of unit circle is 􀀡 ! + 􀀣 ! = 1  \nAs we know cos 􀀩 = 􀀡 and sin 􀀩 = 􀀣 ⇒ cos! 􀀩 + sin! 􀀩 = 1 For all values of 􀀩 ,sin! 􀀩 + cos! 􀀩 ≡ 1  \nFor all values of 􀀩 , such that cos 􀀩 ≠ 0, tan 􀀦 ≡ \\#'$(%\\#&&  \nYou can use the above identities to simplify trignometric expressions and complete proofs  \nExample 4: Simplify 􀁃 . 5 sin! 3􀀾 + 5 cos! 3􀀾 􀁅 . \"($(')(!~~ ~~))  \na. Start by factorising the equation  \n⇒ 5(sin! 3􀀾 + cos! 3􀀾 )  \n⇒ 5 × 1 = 5 As sin! 􀀾 + cos! 􀀾 ≡ 1 ⇒ sin! 3􀀾 + cos! 3􀀾 = 1  \nb.  \n\"($(')(!~~ ~~)) = √&(',-(!~~ ~~.  As cos! 􀀾 + sin! 􀀾 = 1 ⇒ (sin! 􀀾 = 1 − cos! 􀀾)⇒ \"($(')(!~~ ~~)) = (&,'-(.. = tan 􀀾  \nSimple Trignometric equations.  \nIn this section you will learn to solve simple trignometric equations of the form sin 􀀾 = 􀁋, cos 􀀾 = 􀁋 (where −1 ≤ 􀁋 ≤ 1) and tan 􀀾 = 􀁎 (where 􀁎 ∈ ℝ)  \n−1 ≤ 􀁋 ≤ 1 as sin and cos has maximum = 1 and minimum = −","cbCaifSdQ8R6D3cD","https://ap.wps.com/l/cbCaifSdQ8R6D3cD","pdf",742441,"English","# Trigonometric identities cheat sheet\n## Unit circle and angle conventions\n## Signs of sin, cos, tan in each quadrant\n## Converting to acute-angle equivalents\n## Exact values and unit circle equations\n## Using identities to simplify expressions\n## Solving trigonometric equations (simple to harder)","[{\"question\":\"How are simple trigonometric equations solved in this cheat sheet?\",\"answer\":\"Solve equations in forms like sin θ = k, cos θ = k, or tan θ = k within a specified interval. The method includes rearranging to match the target function, finding principal values, and then checking which quadrants produce valid solutions.\"}]","Trigonometric identities - Edexcel Pure Year 1 - Angles and unit circle | PDF",1791914549]