[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-523539-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-523539-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","complex-numbers-and-polar-form-worksheet","Complex Numbers and Polar Form - Worksheet","","Worksheet focused on complex numbers presented in rectangular and polar forms. Students find absolute values, plot complex points on the complex plane using the appropriate coordinate system, convert between rectangular and polar forms, and simplify expressions with complex numbers. Exercises include multiplying complex numbers, converting forms, and identifying and working with nth roots, plus critical thinking items involving roots and solving for a complex variable.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/complex-numbers-and-polar-form-worksheet/523539/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/complex-numbers-and-polar-form-worksheet/523539.png","ImageObject",300,407,{"name":42,"@type":43},"hdkawk","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",15,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you choose between rectangular and polar coordinates when plotting complex numbers?","Question",{"text":63,"@type":64},"Use rectangular coordinates when the complex number is given in rectangular form, and use polar coordinates when polar form is used.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What types of tasks are included for complex numbers in this worksheet?",{"text":68,"@type":64},"The worksheet covers absolute value, plotting points, converting between rectangular and polar form, simplifying products, and finding nth roots.",{"name":70,"@type":61,"acceptedAnswer":71},"What do the critical thinking questions ask you to do?",{"text":72,"@type":64},"They include showing that a complex value is a specific root and solving for a variable in a complex-number equation.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},523539,1791459910,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":124,"preview_page_count":4},3985747859855,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Kuta Software-Infinite Precalculus Name   \nComplex Numbers and Polar Form Date  Period   \nFind the absolute value.  \n1) 􀀗 􀀎 􀀖 i 2) 􀀐 􀀔􀀘 􀀎 i 􀀔􀀘  \n3) 􀀖 (cos ~~ ~~􀀖~~ ~~ 􀀎 isin ~~ ~~􀀖~~ ~~) 4) 􀀕􀀔 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \nPlot each point in the complex plane. Use rectangular coordinates when the number is given in rectangular form and polar coordinates when polar form is used.  \n5) 􀀔 􀀎 􀀖 i  \n6) 􀀖 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎  isin ~~ ~~􀀚􀁓􀀙~~ ~~)  \n􀁓  \n􀀕􀁓   \n􀀖  \n􀀕  \n 􀁓   \n􀀖  \n􀀓  \n􀀗􀁓   􀀘􀁓   \n􀀖 􀀖􀁓 􀀖  \n􀀕  \nConvert numbers in rectangular form to polar form and polar form to rectangular form.  \n7) 􀀐􀀕 􀀖 􀀐 􀀕 i 8) 􀀕􀀔 􀀎 i 􀀚  \n9) 􀀐 ~~ ~~􀀘~~ ~~􀀕~~ ~~􀀕~~ ~~ 􀀐 ~~ ~~􀀘~~ ~~􀀕~~ ~~􀀕~~ ~~i 10) 􀀕 i  \n11) 􀀙 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~) 12) 􀀕(cos 􀀗􀁓􀀖~~ ~~ 􀀎 isin 􀀗􀁓􀀖~~ ~~)  \n13) 􀀖 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~) 14) 􀀖􀀓 (cos 􀁓 􀀎 isin 􀁓)  \nSimplify. Write your answer in rectangular form when rectangular form is given and in polar form when polar form is given.  \n15) (􀀗 􀀎 􀀗i)(􀀘 􀀐 􀀖 i) 16) 􀀗 􀀕 (cos ~~ ~~􀀚􀁓􀀗~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀗~~ ~~) 􀂘 􀀕 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n17) 􀀙􀀕~~ ~~~~ ~~􀀕􀀗ii~~ ~~ 18) 􀀕 􀀙 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀙~~ ~~)  \n􀀙 (cos ~~ ~~􀀔~~ ~~􀀔􀀙􀁓~~ ~~ 􀀎 isin ~~ ~~􀀔~~ ~~􀀔􀀙􀁓~~ ~~)  \n19) (􀀐􀀔 􀀐 􀀙i) 􀀖 20) (􀀕 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀙~~ ~~)) 􀀖  \nFind all nth roots. Write your answers in rectangular form when rectangular form is given and in polar form when polar form is given.  \n21) 􀀕 i, n 􀀠 􀀖 22) 􀀙(cos ~~ ~~􀀖􀁓􀀗~~ ~~ 􀀎 isin ~~ ~~􀀖􀁓􀀗~~ ~~), n 􀀠 􀀗  \nCritical thinking questions:  \n23) Show that 􀀐i 􀀕􀀖 is a 23rd root of ni. 24) Solve for x : (􀀕 􀀎 i)x 􀀠 􀀔 􀀎 􀀕 i Hint: x is a complex number.  \nKuta Software-Infinite Precalculus  \nComplex Numbers and Polar Form  \nFind the absolute value.  \n1) 􀀗 􀀎 􀀖 i  \n􀀘  \n3) 􀀖 (cos ~~ ~~􀀖~~ ~~ 􀀎 isin ~~ ~~􀀖~~ ~~)  \n􀀖  \nName  Date  Period   \n2) 􀀐 􀀔􀀘 􀀎 i 􀀔􀀘  \n􀀖􀀓  \n4) 􀀕􀀔 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n􀀕􀀔  \nPlot each point in the complex plane. Use rectangular coordinates when the number is given in rectangular form and polar coordinates when polar form is used.  \n5) 􀀔 􀀎 􀀖 i  \n6) 􀀖 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎  isin ~~ ~~􀀚􀁓􀀙~~ ~~)  \n􀁓  \n􀀕􀁓   \n􀀖  \n􀀕  \n 􀁓   \n􀀖  \n􀀓  \n􀀗􀁓   􀀘􀁓   \n􀀖 􀀖􀁓 􀀖  \n􀀕  \nConvert numbers in rectangular form to polar form and polar form to rectangular form.  \n7) 􀀐􀀕 􀀖 􀀐 􀀕 i 8) 􀀕􀀔 􀀎 i 􀀚  \n􀀗 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀙~~ ~~) 􀀕 􀀚 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n9) 􀀐  􀀘 􀀕  􀀐  􀀘 􀀕 i 10) 􀀕 i  \n􀀘 ( ~~ ~~􀀘􀁓􀀗~~ ~~ isin ~~ ~~􀀘􀁓􀀗~~ ~~) 􀀕 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n11) 􀀙 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~) 12) 􀀕(cos 􀀗􀁓􀀖~~ ~~ 􀀎 isin 􀀗􀁓􀀖~~ ~~)  \ni 􀀙 􀀐􀀔 􀀐 i 􀀖   \n13)  􀀖 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n􀀖 􀀖  \n􀀎 i  \n􀀕 􀀕  \n14) 􀀖􀀓 (cos 􀁓 􀀎 isin 􀁓)  \n􀀐 􀀖􀀓  \nSimplify. Write your answer in rectangular form when rectangular form is given and in polar form when polar form is given.  \n15) (􀀗􀀖􀀕􀀎􀀎􀀗i􀀛(􀀘 􀀐 􀀖 i) 16) 􀀗 􀀕 (cos ~~ ~~􀀚􀁓􀀗~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀗~~ ~~) 􀂘 􀀕 (cos ~~ ~~~~ ~~ 􀀎 isin ~~ ~~~~ ~~)  \n􀀛 􀀕 (cos ~~ ~~􀀕􀀖􀀔~~ ~~ 􀀎 isin 􀀕􀀖􀀔~~ ~~)  \n17) 􀀙􀀕~~ ~~~~ ~~􀀕􀀗ii~~ ~~ 18) 􀀕 􀀙 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀙~~ ~~)  \n~~ ~~􀀔􀀘 􀀐 ~~ ~~􀀚􀀘~~ ~~i 􀀙 (cos ~~ ~~􀀔~~ ~~􀀔􀀙􀁓~~ ~~ 􀀎 isin ~~ ~~􀀔~~ ~~􀀔􀀙􀁓~~ ~~)  \n~~ ~~􀀖~~ ~~􀀙~~ ~~(cos 􀀐 ~~ ~~􀀕􀁓􀀖~~ ~~ 􀀎 isin 􀀐 ~~ ~~􀀕􀁓􀀖~~ ~~)  \n19) (􀀔􀀓􀀚􀀐􀀎􀀙i􀀔)􀀖􀀜􀀛 i 20) (􀀕 (cos ~~ ~~􀀚􀁓􀀙~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀙~~ ~~)) 􀀖  \n􀀛 (cos ~~ ~~􀀚􀁓􀀕~~ ~~ 􀀎 isin ~~ ~~􀀚􀁓􀀕~~ ~~)  \nFind all nth roots. Write your answers in rectangular form when rectangular form is given and in polar form when polar form is given.  \n21) 􀀕􀀔i􀀑,􀀓 􀀓􀀙􀀖 i 22) 􀀙(cos ~~ ~~􀀖􀁓􀀗~~ ~~ 􀀎 isin ~~ ~~􀀖􀁓􀀗~~ ~~), n 􀀠 􀀗 􀀗􀀙 (cos ~~ ~~􀀖􀀔~~ ~~ 􀀎 isin ~~ ~~􀀖􀀔~~ ~~)  \n􀀐 􀀔􀀔 􀀑􀀑 􀀓􀀕􀀜􀀙 i􀀎 􀀓􀀑􀀙􀀖 i 􀀗􀀗 􀀙􀀙 (coscos ~~ ~~􀀔􀀔􀀔􀁓􀀔􀀙􀀜􀁓􀀔􀀙~~ ~~ 􀀎􀀎 isinisin ~~ ~~􀀔􀀔􀀔􀁓􀀔􀀙􀀜􀁓􀀔􀀙~~ ~~)  \n􀀗􀀙 (cos 􀀕􀀚􀁓􀀔􀀙~~ ~~ 􀀎 isin 􀀕􀀚􀁓􀀔􀀙~~ ~~)  \nCritical thinking questions:  \n23) Show that 􀀐i 􀀕􀀖 is a 23rd root of ni. 24) Solve for x : (􀀕 􀀎 i)x 􀀠 􀀔 􀀎 􀀕 i 􀀗 􀀎  􀀖 i (􀀐i 􀀕􀀖 n ) 􀀕􀀖 = (􀀐i) 􀀕􀀖 􀂘 ( 􀀕􀀖 n ) 􀀕􀀖 = in Hint: x is a complex number. 􀀘 􀀘  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaij45WT28Es29","https://ap.wps.com/l/cbCaij45WT28Es29","pdf",37825,"English","# Find the absolute value\n# Plot points in the complex plane\n# Convert between rectangular and polar form\n# Simplify complex number expressions\n# Find all nth roots\n# Critical thinking questions","[{\"question\":\"How do you choose between rectangular and polar coordinates when plotting complex numbers?\",\"answer\":\"Use rectangular coordinates when the complex number is given in rectangular form, and use polar coordinates when polar form is used.\"},{\"question\":\"What types of tasks are included for complex numbers in this worksheet?\",\"answer\":\"The worksheet covers absolute value, plotting points, converting between rectangular and polar form, simplifying products, and finding nth roots.\"},{\"question\":\"What do the critical thinking questions ask you to do?\",\"answer\":\"They include showing that a complex value is a specific root and solving for a variable in a complex-number equation.\"}]","Complex Numbers and Polar Form - Worksheet | PDF",1791426972]