[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572135-105":53,"doc-detail-572135-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","trigonometric-functions-reciprocal-trigonometric-functions-graphing-identities-and-solving-equations","Trigonometric functions - Reciprocal trigonometric functions - Graphing identities and solving equations","","Reciprocal trigonometric functions are introduced as the functions formed from the reciprocals of the basic trigonometric functions, along with their key domain restrictions where denominators become zero. The document explains how to graph reciprocal functions, state their ranges, and derive related identities. It then guides simplification and identity proving using definitions and identities, emphasizing starting from one side and working systematically to the other. Finally, it extends to solving equations involving reciprocal and inverse trigonometric functions by applying transformations and inverse relationships.",{"@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":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-functions-reciprocal-trigonometric-functions-graphing-identities-and-solving-equations/572135/",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-functions-reciprocal-trigonometric-functions-graphing-identities-and-solving-equations/572135.png","ImageObject",442,249,{"name":88,"@type":89},"xXChrisXx","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"When are reciprocal trigonometric functions undefined, and why?","Question",{"text":108,"@type":109},"They are undefined whenever the original denominator becomes zero. Since division by zero is not defined, the reciprocal functions cannot be evaluated at those x-values.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How are reciprocal trigonometric functions graphed and what key information is given?",{"text":113,"@type":109},"The document instructs sketching the reciprocal functions using domain and range, including restrictions such as x-values that exclude points where the denominator is zero. It also provides ranges, for example sec has y ≤ -1 or y ≥ 1.",{"name":115,"@type":106,"acceptedAnswer":116},"What approach is recommended for proving trigonometric identities?",{"text":117,"@type":109},"Start from one side of the identity and work toward the other side. Use the definitions and identities already covered, and simplify systematically until both sides match.","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},572135,1792107104,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"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":4},8796112263567,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Trigonometric functions Cheat Sheet Reciprocal trigonometric functions  \nPreviously, you have met three trigonometric functions; 􀝏􀝅􀝊􀝔 , 􀜿􀝋􀝏􀝔 􀜽􀝊􀝀 􀝐􀜽􀝊􀝔 .  \nThis chapter introduces three more trigonometric functions, known as the reciprocal trigonometric functions:  \n▪ 􀝏􀝁􀜿􀝔 = 􀯖􀯢1􀯦􀯫 (undefined for values of x for which 􀜿􀝋􀝏􀝔 = 0)  \n▪ 􀜿􀝋􀝏􀝁􀜿􀝔 = 􀯦􀯜1􀯡􀯫 (undefined for values of x for which 􀝏􀝅􀝊􀝔 = 0)  \n▪ 􀜿􀝋􀝐􀝔 = 􀯧􀯔1􀯡􀯫 (undefined for values of x for which 􀝐􀜽􀝊􀝔 = 0)  \nSince division by zero is undefined, we have that these functions are undefined when the denominators are equal to zero.  \nNote that 􀜿􀝋􀝐􀝔 = 􀯧􀯔1􀯡􀯫 = 􀯖􀯦􀯢􀯜􀯦􀯡􀯫􀯫 , simply by replacing 􀝐􀜽􀝊􀝔 with 􀯦􀯖􀯜􀯢􀯡􀯦􀯫􀯫 . This will sometimes be a more useful form to use.  \nCareful: It is not true that: 􀝏􀝁􀜿􀝔 = (􀜿􀝋􀝏􀝔)−1, 􀜿􀝋􀝏􀝁􀜿􀝔 = (􀝏􀝅􀝊􀝔)−1, 􀜿􀝋􀝐􀝔 = (􀝐􀜽􀝊􀝔)−1 The negative power has a different meaning when used with trigonometric functions.  \nGraphing the reciprocal functions  \nYou need to be able to sketch the reciprocal trigonometric functions as well as any transformations, using radians and degrees. Below are the graphs of the reciprocal functions  \n􀝕 = sec 􀝔 Dotted curve represents 􀝕 = 􀜿􀝋􀝏􀝔 Domain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝋􀝀􀝀 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀰗2 (90°)  \nRange: 􀝕 ≤ −1 , 􀝕 ≥ 1  \n􀝕 = 􀜿􀝋􀝏􀝁􀜿􀝔 Dotted curve represents 􀝕 = 􀝏􀝅􀝊􀝔 Domain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀟨 (180°)  \nRange: 􀝕 ≤ −1 􀝋􀝎 􀝕 ≥ 1  \n􀝕 = 􀜿􀝋􀝐􀝔  \nDomain: 􀝔 ∈ ℝ, 􀝔 ≠ 􀝉􀝑􀝈􀝐􀝅􀝌􀝈􀝁 􀝋􀝂 􀟨 (180°)  \nRange: 􀝕 ∈ ℝ  \nThis gives us the following identities:  \n▪ 1 + 􀜿􀝋􀝐 2􀝔 = 􀜿􀝋􀝏􀝁􀜿 2􀝔  \n▪ 1 + 􀝐􀜽􀝊2􀝔 = 􀝏􀝁􀜿 2􀝔  \nYou could be asked to prove these identities, so make sure you are able to reproduce the proofs on the left.  \nSimplifying expressions and proving identities  \nYou can use the definitions and identities we have covered so far to simplify and prove expressions involving the reciprocal trig functions.  \nThere is no trick or standard procedure to be used for these questions. Your ability to manipulate trigonometric expressions using reciprocal functions and identities is being tested, so the most useful thing you can do is properly familiarise yourself with these functions and the above identities. As with most of mathematics, the most useful tool here is practice.  \nWhen proving identities, you must start from one side and work your way towards the other side. You can start from any side, so pick whichever seems like an easier starting point.  \n\n| Example 1: Prove that 􀝏􀝁􀜿 2􀝔 + 􀜿􀝋􀝏􀝁􀜿 2􀝔 = 􀝏􀝁􀜿 2􀝔 􀜿􀝋􀝏􀝁􀜿 2􀝔 |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| Starting from the LHS, we have:\u003Cbr>using the 􀝐􀜽􀝊2􀝔 and 􀝏􀝁􀜿 2􀝔 identities:\u003Cbr>Rewriting 􀝐􀜽􀝊􀝔 as 􀯦􀯖􀯜􀯢􀯡􀯦􀯫􀯫 and 􀜿􀝋􀝐􀝔 as 􀯖􀯦􀯢􀯜􀯦􀯡􀯫􀯫 :\u003Cbr>combining everything into one fraction:\u003Cbr>using 􀝏􀝅􀝊2􀝔 + 􀜿􀝋􀝏 2􀝔 = 1\u003Cbr>splitting the fraction up into a product, giving us the RHS | 􀜮􀜪􀜵 = 􀝏􀝁􀜿 2􀝔 + 􀜿􀝋􀝏􀝁􀜿 2􀝔\u003Cbr>(1 + 􀝐􀜽􀝊2􀝔) + (1 + 􀜿􀝋􀝐 2􀝔) = 2 + 􀝐􀜽􀝊2􀝔 + 􀜿􀝋􀝐 2􀝔 |  |  |  |\n|  | = 2 + | 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 | + | 􀜿􀝋􀝏 2􀝔􀝏􀝅􀝊2􀝔 |\n|  | 􀝏􀝅􀝊4􀝔 + 􀜿􀝋􀝏 4􀝔= 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 + 2\u003Cbr>􀝏􀝅􀝊4􀝔 + 􀜿􀝋􀝏 4􀝔 + 2􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔\u003Cbr>=\u003Cbr>􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔(􀝏􀝅􀝊2􀝔 + 􀜿􀝋􀝏 2􀝔)2   1 \u003Cbr>= =\u003Cbr>􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔 􀝏􀝅􀝊2􀝔􀜿􀝋􀝏 2􀝔\u003Cbr> 1   1 \u003Cbr>= 􀝏􀝅􀝊2􀝔 ∙ 􀜿􀝋􀝏 2􀝔 = 􀝏􀝁􀜿 2􀝔 􀜿􀝋􀝏􀝁􀜿 2􀝔 = 􀜴􀜪􀜵 |  |  |  |\n\nSolving equations  \nPreviously, in Pure Year 1, you learnt how to solve trigonometric equations involving 􀝏􀝅􀝊􀝔, 􀜿􀝋􀝏􀝔 􀜽􀝊􀝀 􀝐􀜽􀝊􀝔 . Now we will look at solving equations that also involve the reciprocal functions. The only difference here is that you need to use the identities and definitions we have covered in this chapter in order to simplify the equation, before you can solve it.  \n\n| Example 2: Solve the equation 􀝏􀝁􀜿􀝔 = ξ2 in the interval 0 ≤ 􀝔 ≤ 360° |  |  |\n| --- | --- | --- |\n| rewriting 􀝏􀝁􀜿􀝔 as 􀯖􀯢1􀯦􀯫 :\u003Cbr>taking the reciprocal of both sides:\u003Cbr>We can solve this via CAST or a graphical method giving: | 1\u003Cbr>􀜿􀝋􀝏􀝔 | = ξ2 |\n|  | ∴ 􀜿􀝋􀝏􀝔 = 1ξ2 􀝔 = 45°, 315° . |  |\n\n\n| Example 3: Solve 􀝏􀝁􀜿 2􀝔 = 3􀝐􀜽􀝊􀝔 in the interval 0 ≤ 􀝔 ≤ 360°. |  |\n| --- | --- |\n| using 1 + 􀝐􀜽􀝊2􀝔 = 􀝏􀝁􀜿 2􀝔\u003Cbr>rearranging\u003Cbr>Solving the quadratic:\u003Cbr>We have two equations to solve. Using CAST ora graphical method, our solutions are: | 1 + 􀝐􀜽􀝊2􀝔 = 3􀝐􀜽􀝊􀝔\u003Cbr>􀝐􀜽􀝊2􀝔 − 3􀝐􀜽􀝊􀝔 + 1 ","cbCaig6cpjFx6mv7","https://ap.wps.com/l/cbCaig6cpjFx6mv7","pdf",973486,"English","# Reciprocal trigonometric functions\n## Definitions and undefined conditions\n## Graphing reciprocal functions\n## Identities\n# Simplifying expressions and proving identities\n# Solving equations with reciprocal functions\n# Inverse trigonometric functions\n## Domain restrictions and reflections\n## Sketching transformations and solving inverse equations","[{\"question\":\"When are reciprocal trigonometric functions undefined, and why?\",\"answer\":\"They are undefined whenever the original denominator becomes zero. Since division by zero is not defined, the reciprocal functions cannot be evaluated at those x-values.\"},{\"question\":\"How are reciprocal trigonometric functions graphed and what key information is given?\",\"answer\":\"The document instructs sketching the reciprocal functions using domain and range, including restrictions such as x-values that exclude points where the denominator is zero. It also provides ranges, for example sec has y ≤ -1 or y ≥ 1.\"},{\"question\":\"What approach is recommended for proving trigonometric identities?\",\"answer\":\"Start from one side of the identity and work toward the other side. Use the definitions and identities already covered, and simplify systematically until both sides match.\"}]","Trigonometric functions - Reciprocal trigonometric functions - Graphing identities and solving equations | PDF",1791925737]