[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-343366-105":59,"doc-detail-343366-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","34-sine-and-cosine-laws-the-sine-and-cosine-rules","3.4 Sine and Cosine Laws - The sine and cosine rules","","A trigonometry lesson explaining the sine rule and cosine rule for finding unknown sides and angles in any triangle, including non-right-angled cases. It defines triangle notation using angles Â, B, C and opposite sides a, b, c, then walks through worked examples using labelled diagrams. The material includes steps for substituting given values into the sine and cosine formulas, solving for angles and lengths, and applying the same trig ideas to triangle area calculations.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/34-sine-and-cosine-laws-the-sine-and-cosine-rules/343366/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/34-sine-and-cosine-laws-the-sine-and-cosine-rules/343366.png","ImageObject",300,407,{"name":92,"@type":93},"Himbo","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem types are the sine and cosine rules used for?","Question",{"text":113,"@type":114},"They help find unknown sides and angles in any triangle, including triangles that are not right-angled.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are sides and angles identified in the rules?",{"text":118,"@type":114},"Angles are labelled Â, B, and C, and the opposite sides are labelled a, b, and c respectively.",{"name":120,"@type":111,"acceptedAnswer":121},"How do you use the cosine rule to find a side length?",{"text":122,"@type":114},"Substitute the given side lengths and included angle into the cosine rule, compute the resulting squared length, then take the square root (rounding as required).","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},343366,1790468236,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},687197100911,"https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697","3.4 Sine and Cosine Laws  \nNovember 20,2014  \n# 3.4 The sine and cosine rules\n\nThe sine and cosine rules are formulaethat will help you to find unknown sidesand angles in a triangle.They enable youto use trigonomentry in triangles thatare not right-angled.  \nThe formula and notation are simpler if youlabel triangles like this.  \nA  \n● The side oppositeA is a.  \na  \nC  \n● The side oppositeB is b.  \n● The side oppositeCis c.  \n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n# →Sine rule\n\nIn any triangle ABC with angles Â,B and C,and oppositesides a,band crespectively:  \nbCB  \nThe sine rule is in theFormula booklet.  \n3.4 Sine and Cosine Laws  \nIn triangle ABC,b=16cm,c=10cm and B=135°.  \na Represent the given information in a labeled diagram.  \nb Find the size of angle C.  \nc Hence find the size of angle A.  \n# Answers\n\n16sin C=10sin 135°C=26.2°(to 3sf)c A+B+C=180°  \nÂ+135°+26.227…=180°Â=18.8°(to 3sf)  \nNovember 20,2014  \n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n|  | →Cosine rule  \u003Cbr>In any triangle ABC with angles A,Band C,  \u003Cbr>and opposite sides a,band crespectively:  \u003Cbr>a²=b²+c²-2bccos   \u003Cbr>This formula can be rearranged to   |\n| --- | --- |\n| These formulae are in  \u003Cbr>the Formula booklet.  \u003Cbr>The first version of  \u003Cbr>the formula is useful  \u003Cbr>when you need to find  \u003Cbr>a side.The second  \u003Cbr>version of the formula  \u003Cbr>is useful when you  \u003Cbr>need to find an angle.   |  |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n| In triangle ABC,AC=8.6m,AB=6.3m and Â=50°.  \u003Cbr>Find the length of BC.  \u003Cbr>Answer  \u003Cbr>Sketch the triangle.  \u003Cbr>BC²=8.6²+6.3²-2×8.6×6.3×cos 50°  \u003Cbr>Use a²=b²+c²  \u003Cbr>BC²=43.9975…  \u003Cbr>BC=6.63m(to 3sf)  \u003Cbr>8.6m  \u003Cbr>50°  \u003Cbr>B  \u003Cbr>A  \u003Cbr>6.3m   |\n| --- |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n| X,Y and Z are three towns.X is 20km due north of Z.  \u003Cbr>Y is to the east of line XZ.The distance from Y to X is 16km and the  \u003Cbr>distance from Z to Y is 8km.  \u003Cbr>a Represent this information in a clear and labeled diagram.  \u003Cbr>b Find the size of angle X.  \u003Cbr>Answers  \u003Cbr>Remember:  \u003Cbr>a       X  \u003Cbr>16km  \u003Cbr>20km  \u003Cbr>8km  \u003Cbr>b  \u003Cbr>动区  \u003Cbr>22.3316  \u003Cbr>=22.3(to 3sf)   |\n| --- |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n| Area   |  of a triangle   |\n| --- | --- |\n|  | Remember that a triangle has three heights,one height per side.  \u003Cbr>base  \u003Cbr>base  \u003Cbr>base   |\n|  |  |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n| Calculate the area of triangle ABC.  \u003Cbr>B  \u003Cbr>7cm  \u003Cbr>42°  \u003Cbr>A  \u003Cbr>10cm  \u003Cbr>C  \u003Cbr>Answer  \u003Cbr>B  \u003Cbr>Use the formula  \u003Cbr>)with ACas the bas,  \u003Cbr>7cm  \u003Cbr>h  \u003Cbr>b=10  \u003Cbr>42°  \u003Cbr>Draw the height,h,the  \u003Cbr>10cm  \u003Cbr>A  \u003Cbr>perpendicular to AC from B.  \u003Cbr>sin  \u003Cbr>Substitute in the formula for the area  \u003Cbr>of a triangle.  \u003Cbr>=23.4cm²(to 3sf)   |\n| --- |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n|  | This formula is in the  \u003Cbr>formula booklet.   |\n| --- | --- |\n| →In any triangle ABC with angles A,Band C,  \u003Cbr>and opposite sides a,b and c respectively,  \u003Cbr>this rule applies:   |  |\n|  |  |\n\n3.4 Sine and Cosine Laws  \nNovember 20,2014  \n\n| Calculate the area of the triangle ABC.  \u003Cbr>8.6m  \u003Cbr>50°  \u003Cbr>A  \u003Cbr>6.3m  \u003Cbr>Answer  \u003Cbr>Substitute in the formula  \u003Cbr>Area of triangle ABC=  \u003Cbr>=20.8m²(to 3sf)   |\n| --- |","cbCaiiHGcbQe4klK","https://ap.wps.com/l/cbCaiiHGcbQe4klK","pdf",3145597,11,"English","# 3.4 The sine and cosine rules\n# Sine rule\n# Cosine rule","[{\"question\":\"What problem types are the sine and cosine rules used for?\",\"answer\":\"They help find unknown sides and angles in any triangle, including triangles that are not right-angled.\"},{\"question\":\"How are sides and angles identified in the rules?\",\"answer\":\"Angles are labelled Â, B, and C, and the opposite sides are labelled a, b, and c respectively.\"},{\"question\":\"How do you use the cosine rule to find a side length?\",\"answer\":\"Substitute the given side lengths and included angle into the cosine rule, compute the resulting squared length, then take the square root (rounding as required).\"}]","3.4 Sine and Cosine Laws - The sine and cosine rules | PDF",1790050920,28]