[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572340-105":53,"doc-detail-572340-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","precalculus-notes-polar-coordinates-polar-equations","Precalculus Notes - Polar Coordinates - Polar Equations","","Precalculus notes covering polar coordinates and polar equations. Explains how a point is represented as an ordered pair (r, θ), definitions of pole and polar axis, and how signs of r and θ affect the coordinates. Provides rules for converting between polar and rectangular forms using x = r cos θ, y = r sin θ, and x^2 + y^2 = r^2, including determining θ via reference angles and quadrants. Includes worked examples transforming equations to circles, plus practice problems.",{"@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/precalculus-notes-polar-coordinates-polar-equations/572340/",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/precalculus-notes-polar-coordinates-polar-equations/572340.png","ImageObject",442,249,{"name":88,"@type":89},"Sophia Brooks","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 is a point defined in polar coordinates (r, θ)?","Question",{"text":108,"@type":109},"A point is defined by an ordered pair (r, θ), where r is the distance from the pole and θ is the angle from the polar axis to the ray reaching the point.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What formulas convert polar coordinates to rectangular coordinates?",{"text":113,"@type":109},"Use x = r cos θ and y = r sin θ, along with x^2 + y^2 = r^2 for substitutions during equation transformations.",{"name":115,"@type":106,"acceptedAnswer":116},"How do you convert rectangular coordinates to polar coordinates?",{"text":117,"@type":109},"Compute r using r = √(x^2 + y^2). Then determine θ using a reference angle and the quadrant; if x = 0, θ is a quadrantal angle.","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},572340,1792059690,{"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":79,"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":73},962084925636,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Precalculus Notes - Polar Coordinates  \n“pole” = origin “polar axis” = ray along positive x axis In Polar Coordinates a point is defined by an ordered pair (r, θ)  \nr: the distance from the point to the pole  \nθ: the angle formed by the polar axis and a ray from the pole to the point (r, θ)  \nExample- Drawing two points in polar coordinates P (2, 􀰗6) and Q (-2, 􀰗6 )  \n1) Draw an angle of 􀰗6 with initial side along the polar axis.  \n2) Plot the point  \na) For point P: go out a distance of 2 along the terminal side of the angle.  \nb) For point Q: go out a distance of 2 along the ray extending in the direction opposite the terminal side of the angle.  \nQ  \nCan we come up with other coordinates for Points P and Q? YES.  \nConsider coordinates with r>0 and r\u003C0 .  \nConsider coordinates with θ>0 and θ\u003C0 .  \nA point with Polar coordinates (r, θ) can also be represented by any of the following: (r, θ + 2πk) or (-r, θ + π + 2πk) where k is any integer  \nP  \nConverting from Polar to Rectangular  \nGiven a point P (r, θ) the rectangular coordinates are x = rcosθ and y = rsinθ(Use “SOH CAH TOA” to derive those formulas)  \nExamples:  \na) (6, 􀰗6) b) (-2, 5􀰗4) c) (-4, -􀰗4)  \nConverting from Rectangular to Polar  \nGiven a point with rectangular coordinates (x, y), find polar coordinates of that point.  \nStep 1: Find r using the formula r = √(􀝔2 + 􀝕 2)  \nStep 2: If x ≠ 0 Use the reference angle α to determine θ: α = 􀝐􀜽􀝊−1| 􀯬􀯫 | After finding α you must consider the quadrant to determine θ . (If x = 0 θ is a quadrantal angle, draw a picture to determine θ) .  \nExamples:  \na) (2, -2) b) (-1, -√3)  \nPrecalculus Notes-Polar Equations  \nWhen transforming equations from Polar to Rectangular form, use the following 3 formulas:  \n(1) x = r cos θ (2) y = r sin θ (3) x2 + y2 = r2  \nThose equations are used to make substitutions. Other helpful steps:  \n􀁸  \n􀁸  \nExample 1)  \nmultiply both sides by “r”(why? to enable us to use formula 3) cross multiply fractions to get rid of denominator  \nr = 4 sin θ  \nStep 1: multiply both sides by r  \nStep 2: Use formula (3) to substitute x2 + y2 for r2  \nStep 3: Use formula (2) to substitute y for r sin θ  \nStep 4: Complete the square in y.  \nResult is a circle with center ( , ) and radius    \nExample 2) 4 x y = 9  \nStep 1: Use formulas (1) and (2) to substitute for x and y  \nStep 2: Simplify and use double angle identity.  \nHW: p. 600 \\#50-64 evens. See back of this sheet for two more examples.  \nConvert from rectangular to polar: 2x2 + 2y2 = 3  \nhint: factor out a 2 on the left hand side and use formula (3)  \nConvert from polar to rectangular: r = cos θ  \nhint: multiply both sides by r, then use formulas (1) and (3), then complete the square","cbCaiaWvUyKNE7rG","https://ap.wps.com/l/cbCaiaWvUyKNE7rG","pdf",405061,"English","# Polar Coordinates\n## Representing a point as (r, θ)\n## Alternative representations using 2πk and sign changes\n# Converting Between Coordinate Systems\n## Polar to Rectangular\n## Rectangular to Polar\n# Polar Equations to Rectangular Form\n## Core substitution formulas\n## Worked example workflows\n# Practice/Homework","[{\"question\":\"How is a point defined in polar coordinates (r, θ)?\",\"answer\":\"A point is defined by an ordered pair (r, θ), where r is the distance from the pole and θ is the angle from the polar axis to the ray reaching the point.\"},{\"question\":\"What formulas convert polar coordinates to rectangular coordinates?\",\"answer\":\"Use x = r cos θ and y = r sin θ, along with x^2 + y^2 = r^2 for substitutions during equation transformations.\"},{\"question\":\"How do you convert rectangular coordinates to polar coordinates?\",\"answer\":\"Compute r using r = √(x^2 + y^2). Then determine θ using a reference angle and the quadrant; if x = 0, θ is a quadrantal angle.\"}]","Precalculus Notes - Polar Coordinates - Polar Equations | PDF",1791926849]