[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-267005-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-267005-en":131},{"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","right-angled-trigonometry-examples-and-formulae","Right-Angled Trigonometry - Examples and Formulae","","Right-angled trigonometry focused on using sine, cosine, and tangent to calculate unknown sides and angles in right-angled triangles. The material explains how to label opposite, adjacent, and hypotenuse, then selects the correct trigonometric ratio for each case. Worked examples include finding lengths such as PQ, XY, BC, AC, and BD, and determining angles using inverse trig. Additional application problems cover elevation and depression in surveyor and cliff scenarios, plus geometry problems involving a pyramid and a prism.",{"@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/right-angled-trigonometry-examples-and-formulae/267005/",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/right-angled-trigonometry-examples-and-formulae/267005.png","ImageObject",300,407,{"name":42,"@type":43},"Mary Man","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-21","2026-09-14",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",16,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you choose between sin, cos, and tan when solving right-angled triangle problems?","Question",{"text":63,"@type":64},"Identify which sides correspond to opposite, adjacent, and hypotenuse. Use sin when you have opposite and hypotenuse, cos when you have adjacent and hypotenuse, and tan when you have opposite and adjacent.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are elevation and depression problems solved in the document?",{"text":68,"@type":64},"Sketch the situation first, then model the geometry using a horizontal distance and an angle of elevation or depression. Apply tangent to relate the distance to the vertical height or depth.",{"name":70,"@type":61,"acceptedAnswer":71},"What method is used for 3D geometry problems like the pyramid and the prism?",{"text":72,"@type":64},"Reduce the 3D problem by taking out relevant 2D cross-sections or shapes. Use Pythagoras in the extracted right triangles, then apply trigonometry to find lengths or angles such as AC, the angle EAH, BD, and angle AD’s relation to a plane.","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},267005,1789413927,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"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":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":111,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":118},7421720224475,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Right-angled Trigonometry  \nSin A 􀀠 Opp CosA 􀀠  Adj  TanA 􀀠 Opp  \nHyp Hyp Adj  \nOne way to remember these formulae is through the mnemonic  \nThe Old Arab Sat On His Camel And Hiccupped  \nExample  \nCalculate the length of PQ in the triangle PQR  \nFirstly label the sides  \nOpp  \nAdj  \nHyp  \nWe are given the ‘hyp’ and require the ‘adj’. Therefore cos  \ncos A 􀀠 AHdjyp PQ  \ncos32 􀀠  \n52  \n52􀁵 cos32 􀀠 PQ  \nPQ 􀀠 44. 1cm  \nExample  \nCalculate the length XY in the triangle XYZ  \nHere we know theAdj and require the Hyp therefore cos  \nHyp  \nOpp  \nAdj  \nAdj  \ncos A 􀀠 ~~  ~~Hyp  \n4.6 cos59 􀀠  \nXY XY 􀁵 cos59 􀀠 4.6  \n 4.6  XY 􀀠  \ncos59  \n􀀠 8.93cm  \nExample  \nCalculate the length of BC in the triangle ABC.  \nDivide the isosceles triangle into two congruent right angled triangles with side 5cm and angle 12⁰[NB an easier way will follow in the following weeks!]  \nx  \nopp  \nsin A 􀀠 oppHypx  \nsin12 􀀠 5  \n5􀁵 sin12 􀀠 xx 􀀠 1.039  \n􀀿 BC = 2 × 1.039 = 2. 08cm  \nExample  \nA surveyor, P, is 1000m away on horizontal ground from the foot of a radio mast, QR. From P the angle of elevation of the top, R, of the mast is 20􀁱 . Find the height of the mast. Assume the surveyor’s height to be negligible.  \nWith a written problem the first step is to draw a sketch.  \n\n|  |  | R |\n| --- | --- | --- |\n| Hyp |  |  |\n|  |  | opp |\n|  20 |  |  |\n\nS  \n1000 PAdj  \ntanA 􀀠 oppAdj  \ntan20 􀀠 1PR000 1000􀁵 tan20 􀀠 PR  \nPR 􀀠 364m  \nExample  \nFrom the top of a vertical cliff 100m high the angle of depression of a boat out at sea is 32 􀁱 . How faris the boat from the foot of the cliff?  \n100  \n\n| | | |\n| --- | --- | --- |\n|  | |  |\n\nd  \n100 tan32 􀀠  \nd  \n 100  d 􀀠  \ntan32 d 􀀠 160.0m  \nExample  \nCalculate the size of angle A in the triangle ABC  \nAdj Opp  \nHyp  \nAs with finding a side when finding an angle we identify which sides will help us. Here we have the ‘opp’ and ‘Hyp’ therefore sin must be used.  \nsin A 􀀠  \nsin A 􀀠  \nopp hyp 4  \n􀀠 0.66666 6  \nA 􀀠 sin 􀀐1(0 .66666)  \nA 􀀠 41.8 􀀤  \nExample  \nFind the size of angle x in each of the following (i)  \ntan x 􀀠  \n1.2  \n4.7  \n􀀠 0.255  \nx 􀀠 tan 􀀐1(0 .255)  \nx 􀀠 14.32 􀀤  \n(ii)  \ntan x 􀀠  \n 10   \n3.7  \n􀀠 0.2.7027  \nx 􀀠 tan 􀀐1(0 .27027)  \nx 􀀠 69.7 􀀤  \n(iii)  \ncos x 􀀠 172 􀀠 x 􀀠 cos 􀀐1  \n0.583 (0 .583)  \nx 􀀠 54.3􀀤  \nExample  \nCalculate the size of angle PQR in the triangle below.  \nP  \n→  \nQ 3  \nSplit the isosceles triangle into two congruent right angled triangles and work with the one with angle Q.  \ncosQ 􀀠 38 􀀠 0.375  \nQ 􀀠 68.0 􀀤  \nExample  \nThe figure is a pyramid on a square base ABCD. The edges of the base are 30cm long and the height, EH, of the pyramid is 42cm. Find  \na) the length of AC  \nb) the angle EAH  \nWith three dimensional problems take out the two dimensional shapes and work with them.  \na)  \nb)  \nA  \n21.2 H  \nPythagoras theorem  \nAC  \n2  \n􀀠 302 􀀎 302  \n􀀠 1800  \nAC 􀀠 1800  \n􀀠 42.43cm  \nAH 􀀠 422.43 􀀠 21.21  42  tanA 􀀠 􀀠 1.97 21.2  \nA 􀀠 63.2 􀀤  \nExample  \nABCDEF is a prism with a triangular cross-section. , BC = 8m, AB = 6m and CD = 20m.  \nFind  \n(i)  \n(ii)  \n(iii)  \n(i)  \n(ii)  \n(iii)  \nAC  \nBD  \nthe angle AD makes with the plane surface BCDF.  \n6  \nA  \nB  \n8 C  \nB F  \n8  \nC  \n6  \nD  \nAC 2 􀀠 62 􀀎 82  \n􀀠 100  \nAC 􀀠 10m  \nBD 2 􀀠 82 􀀎 202  \n􀀠 464  \nBD 􀀠 21.54m  \ntanA 􀀠 21654 􀀠 0.278...  \nA 􀀠 15.56 􀀤  \nD","cbCaiiSIR8E3Arsx","https://ap.wps.com/l/cbCaiiSIR8E3Arsx","pdf",433637,"English","# Right-angled trigonometry formulae\n## Mnemonic and ratio selection\n## Worked examples: sides and angles\n## Applications: elevation, depression, pyramid and prism","[{\"question\":\"How do you choose between sin, cos, and tan when solving right-angled triangle problems?\",\"answer\":\"Identify which sides correspond to opposite, adjacent, and hypotenuse. Use sin when you have opposite and hypotenuse, cos when you have adjacent and hypotenuse, and tan when you have opposite and adjacent.\"},{\"question\":\"How are elevation and depression problems solved in the document?\",\"answer\":\"Sketch the situation first, then model the geometry using a horizontal distance and an angle of elevation or depression. Apply tangent to relate the distance to the vertical height or depth.\"},{\"question\":\"What method is used for 3D geometry problems like the pyramid and the prism?\",\"answer\":\"Reduce the 3D problem by taking out relevant 2D cross-sections or shapes. Use Pythagoras in the extracted right triangles, then apply trigonometry to find lengths or angles such as AC, the angle EAH, BD, and angle AD’s relation to a plane.\"}]","Right-Angled Trigonometry - Examples and Formulae | PDF"]