[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-573879-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-573879-en":126},{"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":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","trigonometric-identities-cheat-sheet","Trigonometric Identities - Cheat Sheet","","A math cheat sheet compiling core trigonometric identities and related formula sets, including reciprocal, quotient, and co-function identities. It also presents parity (even/odd) identities, sum and difference formulas, double-angle and power-reducing/half-angle formulas, and sum-to-product and product-to-sum transformations. Additional sections cover key derivative and integration rules, hyperbolic identities, expression substitution forms, numerical integration methods like trapezoid and Simpson’s rules with error bounds, and arc length and surface area formulas.",{"@graph":14,"@context":72},[15,34,55],{"@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/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/trigonometric-identities-cheat-sheet/573879/",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/trigonometric-identities-cheat-sheet/573879.png","ImageObject",442,249,{"name":42,"@type":43},"pop out","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15","2026-10-14",true,{"@type":52,"interactionType":53,"userInteractionCount":26},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What identity sets are included in the cheat sheet?","Question",{"text":62,"@type":63},"It includes reciprocal identities, quotient identities, co-function identities, parity identities, sum and difference formulas, double-angle formulas, half-angle/power-reducing formulas, and sum-to-product/product-to-sum formulas.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Does the cheat sheet include derivative and integration formulas?",{"text":67,"@type":63},"Yes. It lists derivative formulas and several integration formulas, along with hyperbolic identities and substitution-related expressions.",{"name":69,"@type":60,"acceptedAnswer":70},"Which numerical integration methods and error bounds are provided?",{"text":71,"@type":63},"It provides the Trapezoid Rule and Simpson's Rule, including corresponding error bounds expressed in terms of derivatives.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},573879,1792022861,{"code":4,"msg":81,"data":82},"success",[83,87,92,97,102,107,112,117,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":85,"slug":86},11,90,"presentations",{"id":88,"doc_module":22,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},12,"Resumes",80,"resumes",{"id":93,"doc_module":22,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},14,"Invoices",70,"invoices",{"id":98,"doc_module":22,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},15,"Posters",60,"posters",{"id":103,"doc_module":22,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},16,"Social Media",50,"social-media",{"id":108,"doc_module":22,"doc_module_name":25,"category_name":109,"show_sort_weight":110,"slug":111},17,"Forms",40,"forms",{"id":113,"doc_module":22,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},18,"Letters",30,"letters",{"id":118,"doc_module":22,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},21,"Paper Templates",5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":84,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":26,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":22},962090760408,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","􀀍c 2012 Math Medics LLC. All rights reserved.  \nTRIGONOMETRIC IDENTITIES  \n􀀏 Reciprocal identities  \n􀀏 Pythagorean Identities sin2 u + cos2 u = 1  \n1 + tan2 u = sec2 u  \n1 + cot2 u = csc2 u  \n􀀏 Quotient Identities  \ntan u =  \nsin u  \ncosu  \ncotu =  \ncosu  \nsin u  \n􀀏 Co-Function Identities  \nsin( 􀀙2 􀀀 u) = cos u cos( 􀀙2 􀀀 u) = sin u  \n􀀙   􀀙  \ntan( 􀀀 u) = cotu cot( 􀀀 u) = tan u  \n2 2  \n􀀙   􀀙  \ncsc( 􀀀 u) = sec u sec( 􀀀 u) = csc u  \n2 2  \n􀀏 Parity Identities (Even & Odd)  \nsin(􀀀u) = 􀀀 sin u cos(􀀀u) = cos u tan(􀀀u) = 􀀀 tan u cot(􀀀u) = 􀀀 cot ucsc(􀀀u) = 􀀀 csc u sec(􀀀u) = sec u  \n􀀏 Sum & Di􀀋erence Formulas sin(u 􀀆 v) = sin u cos v 􀀆 cos u sin v cos(u 􀀆 v) = cos u cos v 􀀇 sin u sin v  \n tan u 􀀆 tan v  tan(u 􀀆 v) =  \n1 􀀇 tan utanv  \n􀀏 Double Angle Formulas sin(2u) = 2 sin u cosu cos(2u) = cos2 u 􀀀 sin2 u  \n= 2 cos2 u 􀀀 1  \n= 1 􀀀 2 sin2 u  \n 2 tan u  tan(2u) =  \n1 􀀀 tan2 u  \n􀀏 Power-Reducing/Half Angle Formulas  \nsin2 u = 1 􀀀 cos(2u) 2  \ncos2 u = 1 + cos(2u) 2  \n1 􀀀 cos(2u)  \ntan2 u =  \n1 + cos(2u)  \n􀀏 Sum-to-Product Formulas  \nsin u + sin v = 2 sin 􀀒 u~~ ~~+2~~ ~~v 􀀓 cos 􀀒 u~~ ~~~~ ~~v 􀀓 sin u 􀀀 sin v = 2 cos 􀀒 u~~ ~~+2~~ ~~v 􀀓 sin 􀀒 u~~ ~~~~ ~~v 􀀓 cosu + cosv = 2 cos 􀀒 u~~ ~~+2~~ ~~v 􀀓 cos 􀀒 u~~ ~~~~ ~~v 􀀓 cos u 􀀀 cos v = 􀀀2 sin 􀀒 u~~ ~~+2~~ ~~v 􀀓 sin 􀀒 u~~ ~~~~ ~~v 􀀓  \n􀀏 Product-to-Sum Formulas  \n1  \nsin u sin v = [cos(u 􀀀 v) 􀀀 cos(u + v)] 2  \n1  \ncosu cosv = [cos(u 􀀀 v) + cos(u + v)] 2  \n1  \nsin u cos v = [sin(u + v) + sin(u 􀀀 v)] 2  \n1  \ncos u sin v = [sin(u + v) 􀀀 sin(u 􀀀 v)] 2  \nDerivatives  \ndd((stindan11xx)) = p1~~1~~¡x~~2 ~~ dx  = 1x2  \nd(sec¡dx1~~ ~~x) = xp~~1~~x~~2 ~~¡~~ ~~1 ~~d~~dx (sinhx) = coshx ~~d~~dx (tanhx) = sech2 x~~d~~dx (sechx) = ¡sechx tanhx  \nd(sinh¡1 x)  \ndx d(tanh¡1)x  \ndx d(sech¡1 x)  \ndx  \n=  \n==  \n 1   \np 1+x2  1   \n1¡x2  \n~~ ¡ 1 ~~  \nxp 1¡x2  \nd(cos¡1 x) dx  \nd(cot¡1 x) dx  \nd(csc¡1 x) dx  \n= ¡ ~~ ~~p1~~1~~¡x~~2~~  \n= ¡  ~~ ~~1x2~~ ~~   \n= ¡ xp1x2 ¡ 1  \n~~d~~dx (coshx) = sinhx  \nddxddx ((coth xcsch x))  ¡ccsschch2xxcoth x  \nd(cosh¡1 x) dx  \nd(coth¡1 x) dx  \nd(csch¡1 x) dx  \n=  \n==  \n 1   \npx12 ¡ 1  \n1¡x2   \n¡ 1  \njxjp 1+x2  \nIntegration Formulas  \nRR asxh2xdxdx~~1~~actan¡osh x1 +C C RRR p~~ ~~ax~~ ~~pcos12~~ ~~¡~~1~~x~~2~~hx¡x21dxdxdxsssiinecnh11x CC R sech2 x dx = tanhx + C R csch2 x dx = ¡ cothx + CRRR h~~x~~xx¡x~~2~~x2 nshin¡xhsedx¡1 xch¡;ecC;hx + C RRR h~~x2 ~~¡xx21x dshschx¡1¡1csCC;c; hx + C  \nHyperbolic Identities  \ncoshx = ~~e~~x~~ ~~~~e ~~¡~~ ~~x  \nsinh 2x = 2 sinhx coshx cosh2 x = ~~cosh 2~~2~~x+1 ~~sech¡ 1 x = cosh¡ 1 ~~1~~xcosh2 x ¡ sinh2 x = 1  \nsinhx = ~~e~~x~~ ¡~~2~~e ~~¡~~ ~~x  \ncosh 2x = cosh2 x + sinh2 x sinh2 x = ~~cosh 2~~2~~x¡ 1 ~~csch¡ 1 x = sinh¡ 1 ~~1~~x  \nexpression substitution  \np~~ ~~a2p~~ ~~a2~~ ~~~~ ~~x2x2~~ ~~  xx  a sina tanµµ px2 ¡ a2 x = asec µ  \nTrapezoid RuleRba f (x)dx t ¢2~~x~~ [f(x0 ) + 2f(x1 ) + 2f(x2 ) + ¢ ¢ ¢ + 2f(xn¡ 1 ) + f(xn )] Simpson's Rule  \nRba f (x)dx t ¢3~~x~~ [f(x0 ) + 4f(x1 ) + 2f(x2 ) + 4f(x3 ) + ¢ ¢ ¢ + 2f(xn¡2 ) + 4f(xn¡ 1 ) + f(xn )] Error Bounds  \njET j · K~~ ~~1(b2~~¡~~na2)3~~ ~~ jEM j · K2(b4~~¡~~na2)3~~ ~~ where j f00 (x)j · K  \njES j · K1(b8~~¡~~0na4)5~~ ~~ where j f(4) (x)j · K  \nArc Length and Surface Area  \nL =Rba p 1 + [f0 (x)]2 dx S =Rba 2¼f(x)p 1 + [f0 (x)]2 dx","cbCaiddOFJdu0NoQ","https://ap.wps.com/l/cbCaiddOFJdu0NoQ","pdf",134346,"English","# Trigonometric Identities\n## Reciprocal, Quotient, and Co-Function Identities\n## Parity, Sum & Difference, Double Angle, Half Angle\n## Sum-to-Product and Product-to-Sum\n# Derivatives and Integrals\n## Derivatives\n## Integration Formulas\n# Hyperbolic Identities and Substitution\n## Hyperbolic Identities\n## Expression Substitution\n# Numerical Integration\n## Trapezoid Rule\n## Simpson's Rule and Error Bounds\n# Arc Length and Surface Area","[{\"question\":\"What identity sets are included in the cheat sheet?\",\"answer\":\"It includes reciprocal identities, quotient identities, co-function identities, parity identities, sum and difference formulas, double-angle formulas, half-angle/power-reducing formulas, and sum-to-product/product-to-sum formulas.\"},{\"question\":\"Does the cheat sheet include derivative and integration formulas?\",\"answer\":\"Yes. It lists derivative formulas and several integration formulas, along with hyperbolic identities and substitution-related expressions.\"},{\"question\":\"Which numerical integration methods and error bounds are provided?\",\"answer\":\"It provides the Trapezoid Rule and Simpson's Rule, including corresponding error bounds expressed in terms of derivatives.\"}]","Trigonometric Identities - Cheat Sheet | PDF",1791949988]