[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571378-105":53,"doc-detail-571378-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","trigonometry-reference-sheet-key-formulas-and-identities","Trigonometry Reference Sheet - Key Formulas and Identities","","Trigonometry Reference Sheet presents a compact collection of core relationships for right-triangle and circular-trigonometry. It defines trigonometric ratios (sin, cos, tan) using adjacent, opposite, hypotenuse, and includes reciprocal, cofunction, even/odd, sum-and-difference, double-angle, half-angle, power-reduction, and product-to-sum identities. It also summarizes unit-circle values, common angle tables, graphing parameters for sinusoidal and tangent functions, inverse-function ranges, and laws of sines and cosines for triangle side–angle 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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/trigonometry-reference-sheet-key-formulas-and-identities/571378/",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/trigonometry-reference-sheet-key-formulas-and-identities/571378.png","ImageObject",442,249,{"name":88,"@type":89},"Syukur Setiap Hari","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What trigonometric ratios are defined for adjacent, opposite, and hypotenuse?","Question",{"text":108,"@type":109},"The sheet defines cos, sin, and tan in terms of adjacent x, opposite y, and hypotenuse r, along with reciprocal forms like sec, csc, and cot.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which trig identity categories are included for simplifying expressions?",{"text":113,"@type":109},"It includes cofunction identities, even/odd identities, sum and difference identities, double-angle identities, half-angle identities, power-reduction identities, and product-to-sum and sum-to-product identities.",{"name":115,"@type":106,"acceptedAnswer":116},"How are graphing features and asymptotes described?",{"text":117,"@type":109},"For y = Acos(Bx + C) + D and y = sin(Bx + C) + D it lists amplitude, period, frequency, vertical shift, and phase shift; for y = tan(Bx + C) it notes asymptotes where cos(Bx + C) = 0.","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},571378,1792054397,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"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":9},962093541965,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Trigonometry Reference Sheet  \nDefinitions  \nadjacent x  \n• cos A = ~~ ~~ = ~~  ~~hypotenuse r  \n opposite   y • sinA = =  \nhypotenuse r  \nopposite   y • tanA = =  \nadjacent x  \nhypotenuse r  \n• sec A = ~~ ~~ = ~~  ~~adjacent x  \nhypotenuse  r  \n• csc A = = opposite y  \nadjacent x  \n• cotA = ~~ ~~ = ~~  ~~opposite y  \n\n|  | 0◦ | 30◦ | 45◦ | 60◦ | 90◦ |\n| --- | --- | --- | --- | --- | --- |\n|  | 0 | π 6 | π 4 | π 3 | π 2 |\n\n\n| sin | 0 | 1 2 | √2\u003Cbr>2 | √3\u003Cbr>2 | 1 |\n| --- | --- | --- | --- | --- | --- |\n| cos | 1 | √3\u003Cbr>2 | √2\u003Cbr>2 | 1 2 | 0 |\n| tan | 0 | √3\u003Cbr>3 | 1 | √3 | undefined |\n\n\n| Unit Circle\u003Cbr>􀀀 cos ~~π~~2 , sin ~~π~~2 􀀁 = (0 , 1) |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| \u003Cbr>􀀀 cos 2~~π~~3 , sin 2~~π~~3 􀀁􀀀 cos 3~~π~~4 , sin 3~~π~~4 􀀁 = 􀀐 −\u003Cbr>= 􀀐 − 12 , √23 􀀑\u003Cbr>√22 , √22 􀀑 \u003Cbr>120◦\u003Cbr>􀀀 cos 5~~π~~6 , sin 5~~π~~6 􀀁 = 􀀐 − ~~ ~~√23 , 12 􀀑\u003Cbr>(cos π, sin π) = (−1, 0) \u003Cbr>135◦\u003Cbr>\u003Cbr>150◦\u003Cbr>180◦\u003Cbr>􀀀 cos 7~~π~~6 , sin 7~~π~~6 􀀁 = 􀀐 − ~~ ~~√23 , − 12 􀀑\u003Cbr>210◦\u003Cbr>\u003Cbr>225◦\u003Cbr>􀀀 cos 5~~π~~4 , sin 5~~π~~4 􀀁 = 􀀐 −\u003Cbr>􀀀 cos 4~~π~~3 , sin 4~~π~~3 􀀁\u003Cbr>√22 , − ~~ ~~√22 􀀑 = 􀀐 − 12 , − ~~ ~~√23 􀀑\u003Cbr>240◦\u003Cbr>\u003Cbr>90◦\u003Cbr>270◦\u003Cbr>\u003Cbr>60◦\u003Cbr>􀀀 cos\u003Cbr>~~π~~3 , sin ~~π~~3 􀀁 = 􀀐 12 , √23 􀀑􀀀 cos ~~π~~4 , sin ~~π~~4 􀀁 = 􀀐 √22 , √22 􀀑\u003Cbr>45◦  􀀀 cos ~~π~~6 , sin ~~π~~6 􀀁 = 􀀐 √23 , 12 􀀑 30◦\u003Cbr>0◦  (cos 0 , sin 0) = (1 , 0)\u003Cbr>330◦\u003Cbr>315◦  􀀀 cos 116~~π~~ , sin 116~~π~~ 􀀁 = 􀀐 √23 , −\u003Cbr>300◦\u003Cbr>􀀀 cos\u003Cbr>􀀀 cos 7~~π~~4 , sin 7~~π~~4 􀀁 = 􀀐 √22 , − ~~ ~~√22 􀀑 5~~π~~3 , sin 5~~π~~3 􀀁 = 􀀐 12 , − ~~ ~~√23 􀀑 |  |  | 12 | 􀀑 |\n| 􀀀 cos 3~~π~~2 , sin 3~~π~~2 􀀁 = (0 , −1) |  |  |  |  |\n| Pythagorean Identities | Reciprocal Identities | Ratio Identities |  |  |\n| • cos2 A + sin2 A = 1\u003Cbr>• 1 + tan2 A = sec2 A\u003Cbr>• 1 + cot2 A = csc2 A | • sec A = co1sA\u003Cbr>• csc A = si1nA\u003Cbr>• cotA = ta1n~~ ~~A | • tanA = scinosAA • cotA = csosinAA |  |  |\n\nCofunction Identities  \n• cos 􀀀 ~~π~~2 − A􀀁 = sinA  \n• sin 􀀀 ~~π~~2 − A􀀁 = cos A  \n• tan 􀀀 ~~π~~2 − A􀀁 = cotA  \n• sec 􀀀 ~~π~~2 − A􀀁 = csc A  \n• csc 􀀀 ~~π~~2 − A􀀁 = sec A  \n• cot 􀀀 ~~π~~2 − A􀀁 = tanA  \nEven/Odd Identities  \n• cos(−A) = cos A  \n• sin(−A) = − sinA  \n• tan(−A) = − tanA  \n• sec(−A) = sec A  \n• csc(−A) = − csc A  \n• cot(−A) = − cotA  \nSum and Difference Identities  \n• cos(A ± B) = cos A cos B ∓ sinAsinB  \n• sin(A ± B) = sinA cos B ± cos A sin B  \n• tan(A ± B) = 1ttAan±~~ ~~tanA~~ ~~tanBB  \n\n| Double Angle Identities |  |\n| --- | --- |\n| • cos 2A = cos2 A − sin2 A\u003Cbr>• cos 2A = 2 cos2 A − 1\u003Cbr>• cos 2A = 1 − 2 sin2 A | • sin 2A = 2 cos AsinA\u003Cbr>• tan 2A =  2 tanA  1 − tan2 A |\n| Half Angle Identities\u003Cbr>• cos A2 = ± r ~~ ~~1~~ ~~+~~ ~~c2os~~ ~~A\u003Cbr>• sin A2 = ± r ~~ ~~1~~ ~~−~~ ~~c2os~~ ~~AA 1 − cos A sinA\u003Cbr>• tan ~~ ~~ = ~~ ~~ = ~~  ~~2 sinA 1 + cos A |  |\n| Power Reduction Identities\u003Cbr>• cos2 A = 1~~ ~~+~~ ~~co2s~~ ~~2A\u003Cbr>• sin2 A = 1~~ ~~−~~ ~~co2s~~ ~~2A\u003Cbr>• tan2 A = 11~~ ~~~~ ~~ccooss~~ ~~22AA |  |\n| Sum-to-Product Identities\u003Cbr>• sinA + sin B = 2 sin 􀀒 A~~ ~~+2~~ ~~B 􀀓 cos 􀀒 A~~ ~~−2~~ ~~B 􀀓\u003Cbr>• sinA − sin B = 2 cos 􀀒 A~~ ~~+2~~ ~~B 􀀓 sin 􀀒 A~~ ~~−2~~ ~~B 􀀓\u003Cbr>• cos A + cos B = 2 cos 􀀒 A~~ ~~+2~~ ~~B 􀀓 cos 􀀒 A~~ ~~−2~~ ~~B 􀀓\u003Cbr>• cos A − cos B = −2 sin 􀀒 A~~ ~~+2~~ ~~B 􀀓 sin 􀀒 A~~ ~~−2~~ ~~B 􀀓 |  |\n| Product-to-Sum Identities\u003Cbr>• sinA cos B = 12 [sin(A + B) + sin(A − B)]\u003Cbr>• cos A cos B = 12 [cos(A − B) + cos(A + B)]\u003Cbr>• sinAsinB = 12 [cos(A − B) − cos(A + B)] |  |\n\nSums of Sines and Cosines  \n• A cos x + B sinx = √A2 + B2 sin(x + φ) where cos φ = √A2B+~~ ~~B~~ ~~2 and sin φ = √A2A+~~ ~~B~~ ~~2  \n• A cos x + B sinx = √A2 + B2 cos(x − φ) where cos φ = √A2A+~~ ~~B~~ ~~2 and sin φ = √A2B+~~ ~~B~~ ~~2  \n\n| Circular Sections\u003Cbr>• Arc length: s = rθ\u003Cbr>• Area: A = 12r2 θ\u003Cbr>• Angular velocity: ω = vr |\n| --- |\n| Graphing\u003Cbr>For y = Acos(Bx + C) + D and y = sin(Bx + C) + D ,\u003Cbr>• Amplitude =|A| • Period = 2πB\u003Cbr>• Frequency = B\u003Cbr>• Vertical Shift = D • Phase Shift = −CB\u003Cbr>The graph of y = tan(Bx + C) has asymptotes at the locations where cos(Bx + C) = 0 . |\n| Range of Inverse Functions\u003Cbr>• y = arcco","cbCaihD3Z3SMwPOa","https://ap.wps.com/l/cbCaihD3Z3SMwPOa","pdf",749782,"English","# Trigonometric Definitions\n# Unit Circle and Angle Tables\n## Ratio and Reciprocal Identities\n# Cofunction and Even/Odd Identities\n# Sum, Difference, and Double/ Half-Angle Identities\n# Power Reduction and Product-to-Sum Identities\n# Circular Sections and Graphing\n# Inverse Function Ranges\n# Laws of Sines and Cosines","[{\"question\":\"What trigonometric ratios are defined for adjacent, opposite, and hypotenuse?\",\"answer\":\"The sheet defines cos, sin, and tan in terms of adjacent x, opposite y, and hypotenuse r, along with reciprocal forms like sec, csc, and cot.\"},{\"question\":\"Which trig identity categories are included for simplifying expressions?\",\"answer\":\"It includes cofunction identities, even/odd identities, sum and difference identities, double-angle identities, half-angle identities, power-reduction identities, and product-to-sum and sum-to-product identities.\"},{\"question\":\"How are graphing features and asymptotes described?\",\"answer\":\"For y = Acos(Bx + C) + D and y = sin(Bx + C) + D it lists amplitude, period, frequency, vertical shift, and phase shift; for y = tan(Bx + C) it notes asymptotes where cos(Bx + C) = 0.\"}]","Trigonometry Reference Sheet - Key Formulas and Identities | PDF",1791914547]