[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-397407-105":59,"doc-detail-397407-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","polar-coordinates-cheat-sheet-chapter-5","Polar Coordinates Cheat Sheet - Chapter 5","","Polar Coordinates Cheat Sheet introduces polar coordinates as an alternative way to locate points in two dimensions using distance from the pole and an angle measured anticlockwise from the initial line. It covers how to use (r, θ), conventional sketching conventions, and the conversion relationships between Cartesian and polar forms. The guide then develops key skills for rewriting equations and loci in polar form, sketching common simple and trigonometric polar curves, and identifying loop/range behavior. It concludes with methods for finding enclosed area using polar integration and core tangent considerations.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/polar-coordinates-cheat-sheet-chapter-5/397407/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/polar-coordinates-cheat-sheet-chapter-5/397407.png","ImageObject",300,407,{"name":92,"@type":93},"pop out","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What two measurements define a point in polar coordinates?","Question",{"text":112,"@type":113},"A point in polar coordinates is described by the distance r from the pole (origin) and the angle θ measured anticlockwise between the initial line and the line from the origin to the point.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are Cartesian and polar coordinates related?",{"text":117,"@type":113},"The sheet provides conversion relationships r cos θ = x and r sin θ = y, leading to identities used for converting between forms. It also lists tangent relationships and expressions involving x^2 + y^2.",{"name":119,"@type":110,"acceptedAnswer":120},"How do you find the area enclosed by a polar curve?",{"text":121,"@type":113},"Use polar integration. For a sector enclosed by a polar curve between θ = a and θ = b, apply the formula A = 1/2 ∫ r^2 dθ, and use double-angle cosine identities when integrals involve cos(2θ) forms.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},397407,1790671960,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":8,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":81,"preview_page_count":4},962090760408,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Polar Coordinates Cheat Sheet  \nYou are likely very used to describing the position of a point in two-dimensions using 􀝔 and 􀝕 coordinates. We will now look at an alternative system to describe the position of such a point, known as Polar Coordinates, where we instead use angles and distances to describe positions. An example of where this system is superior to the 􀝔􀝕 system can be found in mechanics; the equation of motion fora particle moving in circular motion is greatly simplified when working with polar coordinates, as opposed to cartesian coordinates.  \nUsing polar coordinates  \nWe need two measurements, 􀝎 and 􀟠 , to describe a point, 􀜲 , in polar coordinates.  \n• 􀝎 is the distance of the point 􀜲 from the pole (the origin 􀜱) .  \n• 􀟠 is the angle measured anticlockwise between the initial line and the line 􀜱􀜲 .  \n􀂃 Polar coordinates are given in the form (􀝎, 􀟠) .  \nNote that when sketching polar curves, it is conventional to draw the lines 􀟠 = 0 and 􀟠 = 􀰗2 instead of the 􀝔 and 􀝕 axes. Keep in mind that not all sketches will have these lines labelled, however.  \nConverting between cartesian and polar forms  \nTo understand the relationship between cartesian and polar coordinates, let’s take a close look at the following diagram:  \nRepresenting an arbitrary point 􀜲 using polar axes.  \nWe can make a few deductions about the relationship between cartesian and polar coordinates from this diagram:  \n􀂃 􀝎 cos 􀟠 = 􀝔 􀂃 􀝎 2 = 􀝔 2 + 􀝕 2  \n⇒  \n􀂃 􀝎 sin 􀟠 = 􀝕 􀂃 tan 􀟠 = 􀯬   \n􀯫  \nYou need to be comfortable using the above four bullet points to convert between the cartesian and polar forms for coordinates and equations.  \nExample 1: Find the polar equation for the curve with cartesian equation 􀝔 2 − 􀝕 2 = 5 .  \n\n| Using 􀝔 = 􀝎 cos 􀟠 , 􀝕 = 􀝎 sin 􀟠 | 􀝎 2 cos2 􀟠 − 􀝎 2 sin2 􀟠 = 5 |\n| --- | --- |\n| Factorise out 􀝎 2 | 􀝎 2 (cos2 􀟠 − sin2 􀟠) = 5 |\n| Use the identity cos 2􀟠 = cos2 􀟠 − sin2 􀟠 | 􀝎 2 (cos 2􀟠) = 5 |\n| Make 􀝎 2 the subject | 􀝎 2 = co5s2􀰏 ∴ 􀝎2 = 5 sec 2􀟠 |\n\nYou can also be expected to convert a locus of points on an Argand diagram into polar form.  \n􀂃 To convert complex loci into polar form, you should first convert into cartesian form, then into polar form.  \nExample 2: Show that the locus of points given by the values of 􀝖 satisfying  \n|􀝖 − 3 − 4􀝅| = 5 can be represented by the polar curve 􀝎 = 6 cos 􀟠 + 8 sin 􀟠 .  \n\n| Convert the locus into cartesian form. (See chapter 2 of Core Pure 1) | This is simply a circle with centre (3,4) and radius 5 ⇒ (􀝔 − 3)2 + (􀝕 − 4)2 = 25 |\n| --- | --- |\n| Use 􀝔 = 􀝎 cos 􀟠 , 􀝕 = 􀝎 sin 􀟠 to now convert into polar form | (􀝎 cos 􀟠 − 3)2 + (􀝎 sin 􀟠 − 4)2 = 25\u003Cbr>􀝎 2 cos2 􀟠 + 􀝎2sin2 􀟠 − 6􀝎 cos 􀟠 − 8􀝎 sin 􀟠 = 25 − 25 |\n| Use sin2 􀟠 + cos2 􀟠 = 1 | 􀝎 2 (1) = 6􀝎 cos 􀟠 + 8􀝎 sin 􀟠 |\n| Divide through by 􀝎 | 􀝎 = 6 cos 􀟠 + 8 sin 􀟠 as required. |\n\nSketching simple polar curves  \nYou need to be able to sketch a small set of simple polar curves, by learning their general shapes.  \nr = a represents a circle with centre 􀜱 and radius 􀜽 .  \n􀝎 = 􀜽􀟠 represents a spiral beginning at 􀜱 .  \n􀟠 = 􀟙 represents a half-line that  \nmakes an angle 􀟙 with the initial line  \nand passes through 􀜱 .  \nSketching harder polar curves  \nYou also need to be able to sketch polar curves involving sine and cosine functions. For such curves, it is a good idea to construct a table of key values of 􀟠 for positive values of 􀝎 to help with your sketch. Here are some facts that are helpful to remember when sketching such curves:  \n􀂃 Equations of the form 􀝎 = 􀜽 cos(􀝊􀟠) or 􀝎 = 􀜽 sin(􀝊􀟠) will have 􀝊 loops symmetrically arranged around the pole (see example 3) .  \n􀂃 Curves of the form 􀝎 = 􀜽(􀝌 + 􀝍 cos 􀟠), where 􀝌 , 􀝍 > 0, will:  \no be circular when 􀝌 = 0 .  \no be a cardioid when p = q.  \no be egg-shaped when 􀯣􀯤 ≥ 2.  \no have a “dimple” when 1 ≤ 􀯣􀯤 \u003C 2 .  \nIf you have a graphical calculator then it is a good idea to use it to help with your sketches.  \nExample 3: Sketch the curve with equation 􀝎 = 􀜽 sin 3􀟠 , where 􀜽 is a positive constant.  \nLooking at the equati","cbCaimAQ9yUvG7t4","https://ap.wps.com/l/cbCaimAQ9yUvG7t4","pdf",501755,"English","# Using polar coordinates\n# Converting between cartesian and polar forms\n# Example conversions and loci\n## Example 1: polar equation from a Cartesian curve\n## Example 2: locus in polar form\n# Sketching simple polar curves\n# Sketching harder polar curves\n## Symmetry and loop behavior\n## Example 3: r = a sin 3θ\n## Example 4: r = a(1 + cos θ)\n# Finding the area enclosed by a polar curve\n# Tangents to polar curves","[{\"question\":\"What two measurements define a point in polar coordinates?\",\"answer\":\"A point in polar coordinates is described by the distance r from the pole (origin) and the angle θ measured anticlockwise between the initial line and the line from the origin to the point.\"},{\"question\":\"How are Cartesian and polar coordinates related?\",\"answer\":\"The sheet provides conversion relationships r cos θ = x and r sin θ = y, leading to identities used for converting between forms. It also lists tangent relationships and expressions involving x^2 + y^2.\"},{\"question\":\"How do you find the area enclosed by a polar curve?\",\"answer\":\"Use polar integration. For a sector enclosed by a polar curve between θ = a and θ = b, apply the formula A = 1/2 ∫ r^2 dθ, and use double-angle cosine identities when integrals involve cos(2θ) forms.\"}]","Polar Coordinates Cheat Sheet - Chapter 5 | PDF",1790461561]