[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572132-105":53,"doc-detail-572132-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","harolds-trigonometry-and-hyperbolic-parent-functions-cheat-sheet-aka-library-of-functions","Harold’s Trigonometry and Hyperbolic Parent Functions Cheat Sheet - AKA Library of Functions","","Cheat sheet compiling core trigonometric and hyperbolic parent functions, plus their inverse versions, in a structured reference format. Covers sine, cosine, tangent, cosecant, secant, and cotangent with domains, ranges, inverse-function notation, restriction notes, and parity (odd/even). Includes arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent, and the hyperbolic set (sinh, cosh, tanh, coth, sech, csch) with inverse identities, domain/range intervals, and asymptote or restriction conditions. Ends with graphing tips.",{"@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":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/harolds-trigonometry-and-hyperbolic-parent-functions-cheat-sheet-aka-library-of-functions/572132/",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/harolds-trigonometry-and-hyperbolic-parent-functions-cheat-sheet-aka-library-of-functions/572132.png","ImageObject",442,249,{"name":88,"@type":89},"miaandmiso","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-25","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What information does the cheat sheet provide for each parent function?","Question",{"text":109,"@type":110},"For each function, it lists domain, range, inverse-function notation, and any restrictions or asymptote notes, along with parity (odd/even). It also includes a general form used for transformations.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How are inverse trigonometric functions presented?",{"text":114,"@type":110},"Each inverse trig function (arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent) includes its own domain and range restrictions, plus a corresponding inverse-function expression and a general form for related transformations.",{"name":116,"@type":107,"acceptedAnswer":117},"Which graphing-related guidance is included at the end?",{"text":118,"@type":110},"The document contains a dedicated section for “Graphing Tips,” intended to support interpreting and sketching the functions after reviewing the domain/range and asymptote information.","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},572132,1792009240,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"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":101,"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":76},7971478544854,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Harold’s  \nTrigonometry and Hyperbolic Parent Functions Cheat Sheet  \nAKA Library of Functions  \n8 January 2026  \n\n| Table of Contents:\u003Cbr>1. Trigonometric\u003Cbr>2. Inverse Trigonometric\u003Cbr>3. Hyperbolic\u003Cbr>4. Inverse Hyperbolic\u003Cbr>5. Graphing Tips |\n| --- |\n| Trigonometric Functions |\n| |\n| Inverse Trigonometric Functions |\n| |\n\n\n| Sine | 􀝂 (􀝔) = 􀝏􀝅􀝊 􀝔 | | Domain: (−∞, ∞) with T = 2π/|b|\u003Cbr>Range: [−1, 1]\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝏􀝅􀝊 −1 􀝔\u003Cbr>Restrictions: None\u003Cbr>Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝏􀝅􀝊 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| --- | --- | --- | --- |\n| Cosine | 􀝂 (􀝔) = 􀜿􀝋􀝏 􀝔 | | Domain: (−∞, ∞) with T = 2π/|b|\u003Cbr>Range: [−1, 1]\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝋􀝏 −1 􀝔\u003Cbr>Restrictions: None Odd/Even: Even General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝋􀝏 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Tangent | 􀝂 (􀝔) = 􀝐􀜽􀝊 􀝔\u003Cbr>􀝏􀝅􀝊 􀝔\u003Cbr>= ~~ ~~􀜿􀝋􀝏 􀝔 | | Domain: (−∞, ∞) except for 􀝔 = 􀰗2 ± 􀝊􀟨\u003Cbr>Range: (−∞, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝐􀜽􀝊 −1 􀝔\u003Cbr>Restrictions: Asymptotes at 􀝔 = 􀰗2 ± 􀝊􀟨 Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝐􀜽􀝊 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Cosecant | 􀝂 (􀝔) = 􀜿􀝏􀜿 􀝔\u003Cbr>1\u003Cbr>= ~~ ~~\u003Cbr>􀝏􀝅􀝊 􀝔 | | Domain: (−∞, ∞) except for 􀝔 = ±􀝊􀟨\u003Cbr>Range: (−∞, -1] ∪ [1, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝏􀜿 −1 􀝔\u003Cbr>Restrictions: Range is bounded Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝏􀜿 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Secant | 􀝂 (􀝔) = sec 􀝔\u003Cbr>1\u003Cbr>= ~~ ~~\u003Cbr>􀜿􀝋􀝏 􀝔 | | Domain: (−∞, ∞) except for 􀝔 = 􀰗2 ± 􀝊􀟨\u003Cbr>Range: (−∞, −1] ∪ [1, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝏􀝁􀜿 −1 􀝔\u003Cbr>Restrictions: Range is bounded Odd/Even: Even\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝏􀝁􀜿 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Cotangent | 􀝂 (􀝔) = 􀜿􀝋􀝐 􀝔\u003Cbr>1\u003Cbr>= ~~ ~~\u003Cbr>􀝐􀜽􀝊 􀝔 | | Domain: (−∞, ∞) except for 􀝔 = ±􀝊􀟨\u003Cbr>Range: (−∞, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝋􀝐 −1 􀝔\u003Cbr>Restrictions: Asymptotes at x = ±􀝊􀟨 Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝋􀝐 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n\nTrigonometric  \nInverse Trigonometric  \n\n| Arcsine | 􀝂 (􀝔) = 􀝏􀝅􀝊 −1 􀝔 | | Domain: [−1, 1]\u003Cbr>Range: [~~ ~~−2􀰗 , 􀰗2] or Quadrants I & IV\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝏􀝅􀝊 􀝔\u003Cbr>Restrictions: Range & Domain are bounded Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝏􀝅􀝊−1 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| --- | --- | --- | --- |\n| Arccosine | 􀝂 (􀝔) = 􀜿􀝋􀝏 −1 􀝔 | | Domain: [−1, 1]\u003Cbr>Range: [0, 􀟨] or Quadrants I & II\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝋􀝏 􀝔\u003Cbr>Restrictions: Range & Domain are bounded Odd/Even: None\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝋􀝏 −1 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Arctangent | 􀝂 (􀝔) = 􀝐􀜽􀝊 −1 􀝔 | | Domain: (−∞, ∞)\u003Cbr>Range: (~~ ~~−2􀰗 , 􀰗2) or Quadrants I & IV\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝐􀜽􀝊 􀝔\u003Cbr>Restrictions: Range is bounded Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝐􀜽􀝊 −1 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Arccosecant | 􀝂 (􀝔) = 􀜿􀝏􀜿 −1 􀝔 | | Domain: (−∞, −1] ∪ [1, ∞)\u003Cbr>Range: [ − 􀰗2 , 0) ∪ (0, 􀰗2] or Quadrants I & IV Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝏􀜿 􀝔 Restrictions: Range & Domain are bounded Odd/Even: Odd\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝏􀜿 −1 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Arcsecant | 􀝂 (􀝔) = 􀝏􀝁􀜿 −1 􀝔 | | Domain: (−∞, −1] ∪ [1, ∞)\u003Cbr>Range: [0, 􀰗2) ∪ (􀰗2 , 􀟨] or Quadrants I & II Inverse Function: 􀝂 −1 (􀝔) = 􀝏􀝁􀜿 􀝔\u003Cbr>Restrictions: Range & Domain are bounded Odd/Even: Neither\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝏􀝁􀜿 −1 (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Arccotangent | 􀝂 (􀝔) = 􀜿􀝋􀝐 −1 􀝔 | | Domain: (−∞, ∞)\u003Cbr>Range: (0, 􀟨) or Quadrants I & II\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝋􀝐 􀝔\u003Cbr>Restrictions: Range is bounded Odd/Even: Neither\u003Cbr>General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝋􀝐 −1 (􀜾 (􀝔 − ℎ)) +􀝇 |\n\n\n| Hyperbolic Graphs |\n| --- |\n| |\n| Inverse Hyperbolic Graphs |\n| |\n\n\n| Hyperbolic Sine | 􀝂 (􀝔) = sinh 􀝔\u003Cbr>􀝁 􀯫 − 􀝁 −􀯫=\u003Cbr>2 | | Domain: (−∞, ∞)\u003Cbr>Range: (−∞, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀝏􀝅􀝊ℎ−1 􀝔\u003Cbr>Restrictions: None Odd/Even: Odd General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀝏􀝅􀝊ℎ (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| --- | --- | --- | --- |\n| Hyperbolic Cosine | 􀝂 (􀝔) = 􀜿􀝋􀝏ℎ 􀝔\u003Cbr>􀝁 􀯫 + 􀝁 −􀯫=\u003Cbr>2 | | Domain: (−∞, ∞)\u003Cbr>Range: [1, ∞)\u003Cbr>Inverse Function: 􀝂 −1 (􀝔) = 􀜿􀝋􀝏ℎ−1 􀝔\u003Cbr>Restrictions: None Odd/Even: Even General Form:\u003Cbr>􀝂 (􀝔) = 􀜽 􀜿􀝋􀝏ℎ (􀜾 (􀝔 − ℎ)) + 􀝇 |\n| Hyperbolic Tangent | 􀝂 (􀝔) = 􀝐􀜽􀝊ℎ 􀝔\u003Cbr>􀝁 2􀯫 − 1 =\u003Cbr>􀝁 2􀯫 + 1 | | Domain: (−∞, ∞)\u003Cbr>Range: (−1, ","cbCairdysSKDyzBB","https://ap.wps.com/l/cbCairdysSKDyzBB","pdf",695232,"English","# Trigonometric\n# Inverse Trigonometric\n# Hyperbolic\n# Inverse Hyperbolic\n# Graphing Tips","[{\"question\":\"What information does the cheat sheet provide for each parent function?\",\"answer\":\"For each function, it lists domain, range, inverse-function notation, and any restrictions or asymptote notes, along with parity (odd/even). It also includes a general form used for transformations.\"},{\"question\":\"How are inverse trigonometric functions presented?\",\"answer\":\"Each inverse trig function (arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent) includes its own domain and range restrictions, plus a corresponding inverse-function expression and a general form for related transformations.\"},{\"question\":\"Which graphing-related guidance is included at the end?\",\"answer\":\"The document contains a dedicated section for “Graphing Tips,” intended to support interpreting and sketching the functions after reviewing the domain/range and asymptote information.\"}]","Harold’s Trigonometry and Hyperbolic Parent Functions Cheat Sheet - AKA Library of Functions | PDF",1791925730]