[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572378-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572378-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","trigonometry-formulas-and-properties","Trigonometry Formulas and Properties","","Trigonometry Formulas and Properties compiles core identities and relationships used to analyze and transform trigonometric expressions. It covers reciprocal, even/odd, and cofunction identities, along with product-to-sum and sum-to-product transformations. Additional sections include sum and difference formulas, periodic identities, and half-angle and double-angle formulas. The reference also provides standard definitions for right-triangle ratios, unit-circle function values, and the Law of Sines, Law of Cosines, and Law of Tangents, plus inverse-trigonometric definitions, domains, ranges, and inverse properties.",{"@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/paper-templates/","Paper Templates",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/trigonometry-formulas-and-properties/572378/",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/trigonometry-formulas-and-properties/572378.png","ImageObject",442,249,{"name":42,"@type":43},"McQueen","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What are reciprocal identities in trigonometry?","Question",{"text":62,"@type":63},"Reciprocal identities relate tangent/cotangent with sine/cosine via csc, sec, and cot. They establish how one trig function can be expressed using another.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How do product-to-sum and sum-to-product formulas help?",{"text":67,"@type":63},"They transform products of sine and cosine terms into sums (and vice versa). This simplifies manipulation and solving trig equations.",{"name":69,"@type":60,"acceptedAnswer":70},"What do the laws of sines and cosines provide?",{"text":71,"@type":63},"The Law of Sines and Law of Cosines relate side lengths and angles in triangles. They enable solving triangles when enough measurements are given.","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},572378,1792061209,{"code":4,"msg":81,"data":82},"success",[83,88,93,98,103,108,113,118,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},11,"Presentations",90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social 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Formulas and Properties  \nReciprocal Identities:  Tangent and Cotangent Identities:  \n| sin 􀟠 = cs1c~~ ~~􀟠 cos 􀟠 = se1c~~ ~~􀟠 tan 􀟠 = co1t~~ ~~􀟠 | csc 􀟠 = si1n~~ ~~􀟠 sec 􀟠 = co1s~~ ~~􀟠 cot 􀟠 = ta1n~~ ~~􀟠 |\n| --- | --- |\n\nEven/Odd Formulas:  \nsin (−􀟠) = − sin 􀟠 csc (−􀟠) = − csc 􀟠  \ncos (−􀟠) = cos 􀟠  \nsec (−􀟠) = sec 􀟠  \ntan (−􀟠) = − tan 􀟠 cot(−􀟠) = − cot 􀟠  \nCofunction Formulas:  \nsin (􀟨2 − 􀟠) = cos 􀟠 cos (􀟨2 − 􀟠) = sin 􀟠  \ncsc (􀟨2 − 􀟠) = sec 􀟠 sec (􀟨2 − 􀟠) = csc 􀟠  \ntan (􀟨2 − 􀟠) = cot 􀟠 cot (􀟨2 − 􀟠) = tan 􀟠  \nProduct to Sum Formulas:  \nsin 􀟙 sin 􀟚 = 12 [cos(􀟙 − 􀟚) − cos(􀟙 + 􀟚)] cos 􀟙 cos 􀟚 = 12 [cos(􀟙 − 􀟚) + cos(􀟙 + 􀟚)] sin 􀟙 cos 􀟚 = 12 [sin(􀟙 + 􀟚) + sin(􀟙 − 􀟚)] cos 􀟙 sin 􀟚 = 12 [sin(􀟙 + 􀟚) − sin(􀟙 − 􀟚)]  \nSum to Product Formulas:  \nsin 􀟙 + sin 􀟚 = 2 sin (􀟙~~ ~~+2~~ ~~􀟚~~ ~~) cos (􀟙~~ ~~−2~~ ~~􀟚~~ ~~ )  \nsin 􀟙 − sin 􀟚 = 2 cos (􀟙~~ ~~+2~~ ~~􀟚~~ ~~ ) sin (􀟙~~ ~~−2~~ ~~􀟚~~ ~~ )  \ncos 􀟙 + cos 􀟚 = 2 cos (􀟙~~ ~~+2~~ ~~􀟚~~ ~~ ) cos (􀟙~~ ~~−2~~ ~~􀟚~~ ~~)  \ncos 􀟙 − cos 􀟚 = −2 sin (􀟙~~ ~~+2~~ ~~􀟚~~ ~~ ) sin (􀟙~~ ~~−2~~ ~~􀟚~~ ~~ )  \nSum and Difference Formulas  \nsin (􀟙 ± 􀟚) = sin 􀟙 cos 􀟚 ± sin 􀟚 cos 􀟙 cos (􀟙 ± 􀟚) = cos 􀟙 cos 􀟚 ∓ sin 􀟙 sin 􀟚  \ntan (􀟙 ± 􀟚) = t1t􀟙a􀟙tatna􀟚n~~ ~~􀟚  \nPeriodic Formulas  \nsin (􀟠 + 2􀟨􀝊) = sin 􀟠 csc (􀟠 + 2􀟨􀝊) = csc 􀟠 cos (􀟠 + 2􀟨􀝊) = cos 􀟠 sec (􀟠 + 2􀟨􀝊) = sec 􀟠 tan(􀟠 + 􀟨􀝊) = tan 􀟠 cot(􀟠 + 􀟨􀝊) = cot 􀟠  \nHalf-Angle Formulas:  \nsin (􀟠2) = ±√1~~ ~~−~~ ~~cos2~~ ~~􀟠  \ncos (􀟠2) = ±√1~~ ~~+~~ ~~cos2~~ ~~􀟠  \ntan (􀟠2) = ±√11~~ ~~~~ ~~coscos~~ ~~􀟠􀟠  \nDouble Angle Formulas:  \nsin (2􀟠) = 2 sin 􀟠 cos 􀟠  \ncos (2􀟠) = cos2 􀟠 −sin2 􀟠= 2 cos2 􀟠 − 1  \n= 1 − 2 sin2 􀟠  \n 2tan 􀟠  tan 2􀟠 =  \n1 − tan2 􀟠  \nTrigonometric Functions:  \nRight Triangle:  Unit Circle:  \nhypotenuse  \nopposite  \nadjacent  \ny  \n(􀝔, 􀝕)  \nr θ  \nx  \nsin θ = hoypppoositetnuse cos θ = haydpjaocetnnutse  \ntan θ =  \nopposite adjacent  \ncsc θ =  \nsec θ =  \nhypotnuse opposite  \nhypotnuse adjacent  \ncot θ =  \nadjacent opposite  \nsin 􀟠 =  \ncos 􀟠 =  \ntan 􀟠 =  \n􀝕  \n􀝎  \n􀝔  \n􀝎  \n􀝕  \n􀝔  \ncsc 􀟠 =  \nsec 􀟠 =  \ncot 􀟠 =  \n􀝎  \n􀝕  \n􀝎  \n􀝔  \n􀝔  \n􀝕  \nLaw of Sines:  \nsin 􀰈  \n􀯔  \n=  \nsin 􀰉  \n􀯕  \n=  \nsin 􀰊  \n􀯖  \nLaw of Cosines:  \n􀜽 2 = 􀜾 2 + 􀜿 2 − 2􀜾􀜿 cos 􀟙  \n􀜾 2 = 􀜽 2 + 􀜿 2 − 2􀜽􀜿 cos 􀟚  \n􀜿 2 = 􀜽 2 + 􀜾 2 − 2􀜽􀜾 cos 􀟛  \nLaw of Tangents:  \n1  \n􀜽 − 􀜾 = tan 2 (􀟙 − 􀟚)  \n􀜽 + 􀜾 tan 1~~2~~ (􀟙 + 􀟚)  \n􀜾 − 􀜿 = tan 1~~2~~ (􀟚 − 􀟛)  \n􀜾 + 􀜿 tan 1~~2~~ (􀟚 + 􀟛)  \n􀜽 − 􀜿 = tan 1~~2~~ (􀟙 − 􀟛)  \n􀜽 + 􀜿 tan 1~~2~~ (􀟙 + 􀟛)  \nInverse Trigonometric Functions  \nDefinition:  \n􀝕 = sin−1 􀝔 ⇔ 􀝔 = sin 􀝕 same as  \n􀝕 = cos −1 􀝔 ⇔ 􀝔 = cos 􀝕 same as  \n􀝕 = tan−1 􀝔 ⇔ 􀝔 = tan 􀝕􀯦􀯔􀯠􀯘 􀯔􀯦  \nAlternative Definition:  \nsin−1 􀝔 =arcsin 􀝔  \ncos −1 􀝔 =arccos 􀝔  \ntan−1 􀝔 =arctan 􀝔  \nDomain and Range:  \nFunction  \n􀝕 = sin−1 􀝔  \n􀝕 = cos −1 􀝔  \n􀝕 = tan−1 􀝔  \nDomain  \n−1 ≤ 􀝔 ≤ 1  \n−1 ≤ 􀝔 ≤ 1  \n−∞ ≤ 􀝔 ≤ ∞  \nRange  \n􀟨  \n− ~~ ~~  \n2  \n≤ 􀝕 ≤  \n􀟨  \n2  \n0 ≤ 􀝕 ≤ 􀟨  \n􀟨  \n− ~~ ~~  \n2  \n\u003C 􀝕 \u003C  \n􀟨  \n2  \n􀝕 = cot −1 􀝔  \n􀝕 = sec −1 􀝔  \n􀝕 = csc −1 􀝔  \n−∞ ≤ 􀝔 ≤ ∞  \n􀝔 ≤ −1, 􀝔 ≥ 1  \n􀝔 ≤ −1, 􀝔 ≥ 1  \n0 ≤ 􀝕 ≤ 􀟨  \n0 ≤ 􀝕 \u003C  \n􀟨  􀟨  \n2 , 2  \n\u003C 􀝕 ≤ 􀟨  \n􀟨  \n−  \n2  \n≤ 􀝕 \u003C 0 , 0 \u003C 􀝕 ≤  \n􀟨  \n2  \nInverse Properties:  \nsin (sin−1(􀝔)) = 􀝔 cos (cos−1(􀝔)) = 􀝔 tan (tan−1(􀝔)) = 􀝔  \nsin−1 (sin(􀟠)) = 􀟠 cos −1 (cos(􀟠)) = 􀟠 tan−1 (tan(􀟠)) = 􀟠","cbCaiaT7Pw3JxRwk","https://ap.wps.com/l/cbCaiaT7Pw3JxRwk","pdf",251156,"English","# Trigonometry Formulas and Properties\n## Reciprocal, Even/Odd, and Cofunction Identities\n## Product-to-Sum and Sum-to-Product Formulas\n## Sum, Difference, Periodic, Half-Angle, and Double-Angle Formulas\n## Trigonometric Functions (Right Triangle and Unit Circle)\n## Law of Sines, Law of Cosines, and Law of Tangents\n## Inverse Trigonometric Functions (Definitions, Domain/Range, Inverse Properties)","[{\"question\":\"What are reciprocal identities in trigonometry?\",\"answer\":\"Reciprocal identities relate tangent/cotangent with sine/cosine via csc, sec, and cot. They establish how one trig function can be expressed using another.\"},{\"question\":\"How do product-to-sum and sum-to-product formulas help?\",\"answer\":\"They transform products of sine and cosine terms into sums (and vice versa). This simplifies manipulation and solving trig equations.\"},{\"question\":\"What do the laws of sines and cosines provide?\",\"answer\":\"The Law of Sines and Law of Cosines relate side lengths and angles in triangles. They enable solving triangles when enough measurements are given.\"}]","Trigonometry Formulas and Properties | PDF",1791927075]