[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-266556-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-266556-en":130},{"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","trig-cheat-sheet-formulas-and-properties","Trig Cheat Sheet - Formulas and Properties","","Trig Cheat Sheet compiles key definitions and core results for trigonometric functions. It starts with right-triangle and unit-circle definitions for sine, cosine, tangent and their reciprocal forms, then summarizes domain, range, and periodicity. It provides a structured set of identities including reciprocal, Pythagorean, even/odd, and periodic forms, plus degree–radian conversion, double-angle, half-angle, and sum/difference formulas. It also includes inverse trigonometric function definitions and properties, followed by laws relating sines, cosines, tangents, and additional angle formulas.",{"@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/trig-cheat-sheet-formulas-and-properties/266556/",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/trig-cheat-sheet-formulas-and-properties/266556.png","ImageObject",300,407,{"name":42,"@type":43},"Aurelia","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-09-20","2026-09-14",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the basic trigonometric definitions for right triangles and the unit circle?","Question",{"text":63,"@type":64},"Sine, cosine, and tangent are defined using opposite/adjacent and hypotenuse ratios in right triangles. The unit-circle definition maps sin(θ) to y, cos(θ) to x, and provides corresponding reciprocal forms.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does the cheat sheet include about domain, range, and period?",{"text":68,"@type":64},"It states that sine and cosine can take any angle, and it lists domain restrictions for other functions. It defines period T using f(θ+T)=f(θ) and lists corresponding periods for each trig function.",{"name":70,"@type":61,"acceptedAnswer":71},"Which identity groups are provided for simplifying expressions?",{"text":72,"@type":64},"The document includes reciprocal, Pythagorean, even/odd, periodic, and cofunction identities. It also provides conversion, double-angle, half-angle, sum/difference, and product-to-sum/sum-to-product formulas.","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},266556,1789408881,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,99,104,109,114,119,122,126],{"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":55,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},6,"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":55,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":80,"read_time":123},1099514068365,"https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068","Definition of the Trig Functions  \nRight triangle definition  \nFor this definition we assume that  \n􀀙 􀀎 􀀎 0 \u003C 􀀒 \u003C 2 or 0 \u003C 􀀒 \u003C 90 .  \nsin (􀀒) = hoyppopotesnituesecos (􀀒) = haydpjacentotenuse  \nopposite tan (􀀒) =  \nadjacent  \ncsc (􀀒) = hyopotepponsiutsee sec (􀀒) = hyapdotenusjacentecot (􀀒) = adopjaceposintte  \nUnit Circle Definition  \nFor this definition 􀀒 is any angle.  \nsin (􀀒) = y1 = y  \ncos (􀀒) = x1 = xy  \ntan (􀀒) = x  \ncsc (􀀒) = 1y 1  \nsec (􀀒) = xx  \ncot (􀀒) = y  \nFacts and Properties  \nDomain  \nThe domain is all the values of 􀀒 that can be plugged into the function.  \nsin (􀀒) , 􀀒 can be any angle cos (􀀒) , 􀀒 can be any angle  \ntan (􀀒) , 􀀒  􀀒n + 12􀀓 􀀙; n = 0 ; 􀀆1; 􀀆2; : : : csc (􀀒) , 􀀒  n􀀙; n = 0 ; 􀀆1; 􀀆2; : : :  \nsec (􀀒) , 􀀒  􀀒n + 12􀀓 􀀙; n = 0 ; 􀀆1; 􀀆2; : : : cot (􀀒) , 􀀒  n􀀙; n = 0 ; 􀀆1; 􀀆2; : : :  \nRange  \nPeriod  \nThe period of a function is the number, T , such that f (􀀒 + T ) = f (􀀒) . So, if ! is a fixed number and 􀀒 is any angle we have the following periods.  \nsin (! 􀀒) ! T = 2􀀙!  \n2􀀙  \ncos (! 􀀒) ! T =  \n!  \n􀀙  \ntan (! 􀀒) ! T = csc (! 􀀒) ! T = 2!􀀙!  \n2􀀙  \nsec (! 􀀒) ! T =  \n!  \n􀀙  \ncot (! 􀀒) ! T =  \n!  \nThe range is all possible values to get out of the function.  \n􀀀1 􀀔 sin (􀀒) 􀀔 1 􀀀1 􀀔 cos (􀀒) 􀀔 1  \n􀀀1 \u003C tan(􀀒) \u003C 1 􀀀1 \u003C cot (􀀒) \u003C 1 sec (􀀒) 􀀕 1 and sec (􀀒) 􀀔 􀀀1 csc (􀀒) 􀀕 1 and csc (􀀒) 􀀔 􀀀1  \n© October 2025 Paul Dawkins-[https://tutorial.math.lamar.edu](https://tutorial.math.lamar.edu)  \nFormulas and Identities  \nTangent and Cotangent Identities  \n sin (􀀒)   cos (􀀒)  \ntan (􀀒) = cot (􀀒) =  \ncos (􀀒) sin (􀀒)  \nReciprocal Identities  \ncsc (􀀒) = sin1(􀀒)  1   \nsec (􀀒) = cos (􀀒)  1   \ncot (􀀒) = tan (􀀒)  \nsin (􀀒) = cs1c(􀀒)  1   \ncos (􀀒) = sec (􀀒)  1   \ntan (􀀒) = cot (􀀒)  \nPythagorean Identities  \nsin2 (􀀒) + cos2 (􀀒) = 1  \ntan2 (􀀒) + 1 = sec2 (􀀒)  \n1 + cot2 (􀀒) = csc2 (􀀒)  \nEven/Odd Formulas  \nsin (􀀀􀀒) = 􀀀 sin(􀀒) csc (􀀀􀀒) = 􀀀 csc(􀀒)  \ncos (􀀀􀀒) = cos(􀀒) sec (􀀀􀀒) = sec(􀀒)  \ntan (􀀀􀀒) = 􀀀 tan(􀀒) cot (􀀀􀀒) = 􀀀 cot(􀀒)  \nPeriodic Formulas  \nIf n is an integer then,  \nsin (􀀒 + 2􀀙n) = sin(􀀒) csc (􀀒 + 2􀀙n) = csc(􀀒)  \ncos (􀀒 + 2􀀙n) = cos(􀀒) sec (􀀒 + 2􀀙n) = sec(􀀒)  \ntan (􀀒 + 􀀙n) = tan(􀀒) cot (􀀒 + 􀀙n) = cot(􀀒)  \nDegrees to Radians Formulas  \nIf x is an angle in degrees and t is an angle in radians then  \n1􀀙80 = tx ) t = 􀀙x180 and x = 180􀀙t  \nDouble Angle Formulas  \nsin(2􀀒) = 2 sin(􀀒) cos (􀀒)  \ncos(2􀀒) = cos2 (􀀒) 􀀀 sin2 (􀀒)  \n= 2 cos2 (􀀒) 􀀀 1  \n= 1 􀀀 2 sin2 (􀀒)  \n 2 tan(􀀒)  tan(2􀀒) =  \n1 􀀀 tan2 (􀀒)  \nHalf Angle Formulas  \nsin 􀀒 􀀒2􀀓 = 􀀆 r ~~ ~~1~~ ~~􀀀~~ ~~co2s~~ ~~(􀀒)   \ncos 􀀒 􀀒2􀀓 = 􀀆 r ~~ ~~1~~ ~~+~~ ~~co2s~~ ~~(􀀒)  \ntan 􀀒 􀀒2􀀓 = 􀀆 s ~~ ~~11~~ ~~~~ ~~coscos((􀀒􀀒))  \nHalf Angle Formulas (alternate form)  \nsin2 (􀀒) = ~~1~~2 (1 􀀀 cos(2􀀒))  \ncos2 (􀀒) = ~~1~~2 (1 + cos(2􀀒))  \ntan2 (􀀒) = 1 􀀀 cos(2􀀒)  \n1 + cos(2􀀒)  \nSum and Difference Formulas  \nsin (􀀋 􀀆 􀀌) = sin(􀀋) cos (􀀌) 􀀆 cos(􀀋) sin (􀀌)  \ncos (􀀋 􀀆 􀀌) = cos(􀀋) cos (􀀌) 􀀇 sin(􀀋) sin (􀀌)  \n tan (􀀋) 􀀆 tan(􀀌)  tan (􀀋 􀀆 􀀌) =  \n1 􀀇 tan(􀀋) tan (􀀌)  \nProduct to Sum Formulas  \nsin (􀀋) sin (􀀌) = ~~1~~2 [cos(􀀋 􀀀 􀀌) 􀀀 cos(􀀋 + 􀀌)] cos (􀀋) cos (􀀌) = ~~1~~2 [cos(􀀋 􀀀 􀀌) + cos(􀀋 + 􀀌)] sin (􀀋) cos (􀀌) = ~~1~~2 [sin(􀀋 + 􀀌) + sin(􀀋 􀀀 􀀌)] cos (􀀋) sin (􀀌) = ~~1~~2 [sin(􀀋 + 􀀌) 􀀀 sin(􀀋 􀀀 􀀌)]  \nSum to Product Formulas  \nsin (􀀋) + sin(􀀌) = 2 sin 􀀒 􀀋~~ ~~+2~~ ~~􀀌 􀀓 cos 􀀒 􀀋~~ ~~~~ ~~􀀌 􀀓 sin (􀀋) 􀀀 sin(􀀌) = 2 cos 􀀒 􀀋~~ ~~+2~~ ~~􀀌 􀀓 sin 􀀒 􀀋~~ ~~~~ ~~􀀌 􀀓 cos (􀀋) + cos(􀀌) = 2 cos 􀀒 􀀋~~ ~~+2~~ ~~􀀌 􀀓 cos 􀀒 􀀋~~ ~~~~ ~~􀀌 􀀓 cos (􀀋)􀀀cos(􀀌) = 􀀀2 sin 􀀒 􀀋~~ ~~+2~~ ~~􀀌 􀀓 sin 􀀒 􀀋~~ ~~~~ ~~􀀌 􀀓  \nCofunction Formulas  \nsin 􀀐 􀀙2 􀀀 􀀒􀀑 = cos (􀀒) cos 􀀐 􀀙2 􀀀 􀀒􀀑 = sin (􀀒)  \ncsc 􀀐 􀀙2 􀀀 􀀒􀀑 = sec (􀀒) sec 􀀐 􀀙2 􀀀 􀀒􀀑 = csc (􀀒)  \ntan 􀀐 􀀙2 􀀀 􀀒􀀑 = cot (􀀒) cot 􀀐 􀀙2 􀀀 􀀒􀀑 = tan (􀀒)  \n© October 2025 Paul Dawkins-[https://tutorial.math.lamar.edu](https://tutorial.math.lamar.edu)  \nFor any ordered pair on the unit circle (x; y) : cos (􀀒) = x and sin(􀀒) = y  \nExample  \ncos 􀀒 5􀀙3 􀀓 = 12 sin 􀀒 5􀀙3 􀀓 = 􀀀 p23  \n© October 2025 Paul Dawkins-[https://tutorial.math.lamar.edu](https://tutorial.math.lamar.edu)  \nInverse Trig Functions  \nDefinition  \ny = sin􀀀1(x) is equivalen","cbCaibN3OOHT5Dcc","https://ap.wps.com/l/cbCaibN3OOHT5Dcc","pdf",189457,"English","# Definitions\n## Right-triangle definitions\n## Unit circle definitions\n# Facts and properties\n## Domain and range\n## Period\n# Formulas and identities\n## Tangent and cotangent identities\n## Reciprocal identities\n## Pythagorean identities\n## Even/odd formulas\n## Periodic formulas\n## Degrees to radians\n## Double-angle and half-angle formulas\n## Sum and difference formulas\n## Product-to-sum and sum-to-product\n## Cofunction formulas\n# Inverse trig functions\n## Definitions and inverse properties\n## Alternate notation\n# Laws and additional formulas","[{\"question\":\"What are the basic trigonometric definitions for right triangles and the unit circle?\",\"answer\":\"Sine, cosine, and tangent are defined using opposite/adjacent and hypotenuse ratios in right triangles. The unit-circle definition maps sin(θ) to y, cos(θ) to x, and provides corresponding reciprocal forms.\"},{\"question\":\"What does the cheat sheet include about domain, range, and period?\",\"answer\":\"It states that sine and cosine can take any angle, and it lists domain restrictions for other functions. It defines period T using f(θ+T)=f(θ) and lists corresponding periods for each trig function.\"},{\"question\":\"Which identity groups are provided for simplifying expressions?\",\"answer\":\"The document includes reciprocal, Pythagorean, even/odd, periodic, and cofunction identities. It also provides conversion, double-angle, half-angle, sum/difference, and product-to-sum/sum-to-product formulas.\"}]","Trig Cheat Sheet - Formulas and Properties | PDF"]