[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572284-105":53,"doc-detail-572284-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","polar-coordinates-cheat-sheet-aqa-a-level-further-maths-core","Polar Coordinates Cheat Sheet AQA A Level Further Maths - Core","","Polar coordinates describe each point using radial distance from the origin (pole) and an angle from the initial line. The cheat sheet explains how to represent points, rewrite equations in polar form, and sketch more complex polar curves by analyzing key values like maxima, zeros, and periodicity. It also derives and applies the A-level area formula for regions enclosed by polar curves, including worked examples using trigonometric identities and integration to find enclosed area and parameter values.",{"@graph":63,"@context":119},[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":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/polar-coordinates-cheat-sheet-aqa-a-level-further-maths-core/572284/",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/polar-coordinates-cheat-sheet-aqa-a-level-further-maths-core/572284.png","ImageObject",442,249,{"name":88,"@type":89},"dawugda","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-21","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How are points represented in polar coordinates?","Question",{"text":109,"@type":110},"A point P is written as (r, θ), where r is the radial distance from the origin (pole) and θ is the angle between the radial line to the point and the x-axis (initial line).","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How do you rewrite an equation into polar coordinates?",{"text":114,"@type":110},"Substitute x = r cos θ and y = r sin θ (and use r = √(x² + y²) and θ = arctan(y/x) when needed) to convert the given relation into r and θ form.",{"name":116,"@type":107,"acceptedAnswer":117},"What formula finds the area enclosed by a polar curve?",{"text":118,"@type":110},"For the enclosed region, use A = (1/2) ∫ r^2 dθ, evaluated between the appropriate θ-limits for the curves and boundary rays.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572284,1792033453,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"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":4},3985747870947,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Polar Coordinates Cheat Sheet AQA A Level Further Maths: Core  \nThe familiar 􀝔 and 􀝕 axes of the 2D plane are just one set of coordinates which can be used to describe each point in the plane. Another set which could be used are called polar coordinates where each point is described by its radial distance from the origin and an angle.  \nRepresenting a Point  \nAny point P can be described by the coordinates (􀝎, 􀟠), where 􀝎 is the radial distance from the point to the origin (also referred to as the pole) and 􀟠 is the angle subtended by the radial line connecting the point to the origin and the x-axis. Note that the 􀝔 axis is often called the initial line.  \nTrigonometry and the Pythagorean theorem can be used to relate 􀝎 and 􀟠 to 􀝔 and 􀝕 in order to convert between coordinate systems:  \nSimple Polar Curves  \nThe simplest possible harder polar curves.  \npolar curves  \nare shown below. These provide a good starting  \nthinking about  \npoint for  \nExample 3: Find  \nthe area 􀜣 , bounded by the half lines 􀟠 = 􀰗3  \n, 􀟠 = 􀰗2  \nand the curves 􀝎 = 2 cos 􀟠  \nand 􀝎 = sin 2􀟠 .  \nBegin by noting the area asked for is the  \ndifference between the areas enclosed by 􀜣 = 􀜣􀯕􀯜􀯚 − 􀜣􀯦􀯠􀯔􀯟􀯟  \neach curve and the half lines.  \n\n|  |  |  |  |\n| --- | --- | --- | --- |\n| Proceed by finding 􀜣􀯕􀯜􀯚 , the area enclosed by the curve 􀝎 = 2 cos 􀟠 and the half lines. | 􀜣􀯕􀯜􀯚 = 12 | 􀰗\u003Cbr>∫􀰗2 4 cos2 􀟠 􀝀􀟠 3 |  |\n| Using the identity re-arranged from earlier cos2 􀟠 = 12 (1 + cos 2􀟠), the integral can be evaluated. | 􀰗\u003Cbr>􀜣􀯕􀯜􀯚 = ∫􀰗21 + cos 2􀟠 􀝀􀟠 3\u003Cbr>⟹ 􀜣􀯕􀯜􀯚 = (􀟠 + 12 sin 2􀟠) |􀰗􀰗23\u003Cbr>􀟨 √2 = −\u003Cbr>6 2 |  |  |\n| 􀜣􀯦􀯠􀯔􀯟􀯟 is similarly found. | 􀜣􀯦􀯠􀯔􀯟􀯟 = 12 |  | ∫􀰗􀰗⁄2sin22􀟠 􀝀􀟠 3 |\n| The other identity can be used to evaluate the integral. | 􀜣􀯦􀯠􀯔􀯟􀯟 = 14\u003Cbr>􀰗\u003Cbr>∫􀰗21 − cos 4􀟠􀝀􀟠 3\u003Cbr>= 14 (􀟠 − 14 sin 4􀟠) |􀰗􀰗23\u003Cbr>􀟨 √3 = −\u003Cbr>24 32 |  |  |\n| 􀜣 can now be found as the difference. | \u003Cbr>􀜣 = 􀟨8 − 7√323 |  |  |\n\nExample 1: Rewrite the equation 􀜽􀝔 − 􀜾􀝕 = 0 in polar coordinates.  \n\n| Directly substituting in the equations for 􀝔 and 􀝕 . | 􀜽 cos 􀟠 − 􀜾 sin 􀟠 = 0 |\n| --- | --- |\n| Rearranging for a simpler form. | tan 􀟠 = ab |\n\n􀝕  \n􀝔  \n􀝎 = √􀝔2 + 􀝕 2  \nand 􀟠 = arctan (􀯬􀯫  \n)  \n􀝔 = 􀝎 cos 􀟠 and 􀝕 = 􀝎 sin 􀟠  \nA circle of radius 􀜽 is given by 􀝎 = 􀜽  \nA spiral is given by 􀝎 = 􀜽􀟠 A half-line is given by θ = a  \nSketching Complex Polar Curves  \nComplex polar curves can be difficult to intuit. Given a function 􀝎 = 􀝎 (􀟠), the general shape of this curve can be investigated by looking at key points such as when 􀝎 is maximum or zero.  \nThere are two common examples of complex curves worth remembering the general shape of:  \n• 􀝎 = 􀜽 cos 􀝊􀟠 or 􀝎 = 􀜽 sin 􀝊􀟠 . If 􀝊 is odd, then 􀝊 loops are produced, if 􀝊 is even then 2􀝊 loops are produced.  \n• 􀝎 = 􀜽 (􀜾 + cos 􀟠) . This is best analysed on a case-by-case basis:  \no If |􀜾| ≥ 2, then it produces an egg shape.  \no If 1 \u003C |􀜾| \u003C 2, then it produces an egg shape with a dimple on one side.  \no If |b| = 1, then it produces a cardioid (a heart shape curved) .  \n􀝎 = 1 + cos 􀟠 , this is a cardioid.  \nExample 2: Sketch the  \ncurve given by 􀝎 =  \nsin 3􀟠 .  \nFirst, by noting that this is periodic every 2􀰗3  \n, the amount of  \nwork needed can be reduced as only the key elements of the curve between 􀟠 = 0 and 􀟠 = 2􀰗3  \nare required to sketch  \nthe full curve. These are:  \n􀟨  \n􀟠: 0  \n6  \n\n|  |  | | 􀟨\u003Cbr>3 | 􀟨\u003Cbr>2 | 2􀟨 3 |\n| --- | --- | --- | --- | --- | --- |\n| 􀝎: | 0 | 1 | 0 | -1 | 0 |\n\nThese points alone produce 2 loops, by rotating the pattern round and accounting for overlap, the desired 3 loop pattern is obtained.  \n􀝎 = 2.1 + cos 􀟠 , this is eggshaped.  \n􀝎 = 1.5 + cos 􀟠 , this is with a dimple.  \negg-shaped  \nWhen analysing an equation of the above form cos (􀟠 − π2  \nbut with sin 􀟠 instead of cos 􀟠 we can  \nuse the identity  \n) = sin 􀟠 to see that it gives the same curve just rotated anti-clockwise by 􀰗2  \n. This is exactly analogous as to how 􀝕 = 􀝂 (􀝔 − 􀜽) is just 􀝕 = 􀝂 (􀝔) shifted by 􀜽 in the 􀝔 direction.  \nArea Enclosed by a Polar Curve (A Level Only)  \nIn cartesian ","cbCaimbeywpmEoRU","https://ap.wps.com/l/cbCaimbeywpmEoRU","pdf",524796,"English","# Representing a Point\n# Converting Between Coordinate Systems\n# Simple Polar Curves\n# Sketching Complex Polar Curves\n# Area Enclosed by a Polar Curve (A Level Only)","[{\"question\":\"How are points represented in polar coordinates?\",\"answer\":\"A point P is written as (r, θ), where r is the radial distance from the origin (pole) and θ is the angle between the radial line to the point and the x-axis (initial line).\"},{\"question\":\"How do you rewrite an equation into polar coordinates?\",\"answer\":\"Substitute x = r cos θ and y = r sin θ (and use r = √(x² + y²) and θ = arctan(y/x) when needed) to convert the given relation into r and θ form.\"},{\"question\":\"What formula finds the area enclosed by a polar curve?\",\"answer\":\"For the enclosed region, use A = (1/2) ∫ r^2 dθ, evaluated between the appropriate θ-limits for the curves and boundary rays.\"}]","Polar Coordinates Cheat Sheet AQA A Level Further Maths - Core | PDF",1791926567]