[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572375-105":53,"doc-detail-572375-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","harolds-triangles-cheat-sheet-trig-laws-and-formulas","Harold’s Triangles Cheat Sheet - Trig Laws and Formulas","","Concise reference covering key triangle-trigonometry relationships and formulas for solving angles, sides, and area. Summarizes Law of Sines, Law of Cosines, Law of Tangents, Law of Cotangents, Pythagorean Theorem, angle-sum identity, area and semi-perimeter formulas, plus Heron’s and Mollweide’s formulas. Includes an order-of-use guide, case breakdown for SSS/SAS/SSA-style inputs, ambiguous SSA solution counts, fixed-angle triangle results, Pythagorean triples, and unit-circle notes on radians and arc length.",{"@graph":63,"@context":119},[64,80,102],{"@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":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-triangles-cheat-sheet-trig-laws-and-formulas/572375/",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/harolds-triangles-cheat-sheet-trig-laws-and-formulas/572375.png","ImageObject",442,249,{"name":88,"@type":89},"Stanley","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",9,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"Which formula should be used first when solving a triangle with angles and sides?","Question",{"text":109,"@type":110},"Start with the Sum of Angles. Use the Pythagorean Theorem only for right triangles, then consider Law of Sines before Law of Cosines when possible, reserving Law of Cosines for cases where it is needed.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What do you do in an SSA ambiguous case?",{"text":114,"@type":110},"Determine the number of solutions based on relative side lengths (typically comparing a and b) and the height term h = a sin(θ). The document notes outcomes ranging from 0 to 2 solutions, including one solution when a = b.",{"name":116,"@type":107,"acceptedAnswer":117},"How are radians related to arc length on a unit circle?",{"text":118,"@type":110},"The cheat sheet states that radians equal arc length s on a unit circle, with s = rθ and C = 2πr. It also provides conversions between degrees and radians using multiples of π.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572375,1792002419,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":73},2336477405376,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Harold’s Triangles Cheat Sheet  \n22 September 2026  \nTrig Laws and Formulas  \n\n| Law | Equation |  |\n| --- | --- | --- |\n| Reference Triangle | |  |\n| Law of Sines | 􀢙􀢏􀢔 (􀡭) 􀢙􀢏􀢔 (􀡮) 􀢙􀢏􀢔 (􀡯) ~~ ~~ = ~~ ~~ = ~~ ~~􀢇 􀢈 􀢉 | 􀜽 􀜾 􀜿 ~~ ~~ = ~~ ~~ = ~~ ~~􀝏􀝅􀝊 (􀜣) 􀝏􀝅􀝊 (􀜤) 􀝏􀝅􀝊 (􀜥) |\n| Law of Cosines | 􀜽 2 = 􀜾 2 + 􀜿 2 − 2􀜾􀜿 ∙ 􀜿􀝋􀝏 (􀜣)\u003Cbr>􀜾 2 = 􀜽 2 + 􀜿 2 − 2􀜽􀜿 ∙ 􀜿􀝋􀝏 (􀜤)\u003Cbr>􀢉􀫛 = 􀢇􀫛 + 􀢈􀫛 − 􀫛􀢇􀢈 ∙ 􀢉􀢕􀢙 (􀡯) |  |\n| Law of Tangents | 􀜽 − 􀜾   􀝐􀜽􀝊 [(􀜣~~ ~~−~~2 ~~􀜤)]=\u003Cbr>􀜽 + 􀜾 􀝐􀜽􀝊 [(􀜣~~ ~~+~~2 ~~􀜤)]\u003Cbr>􀜾 − 􀜿   􀝐􀜽􀝊 [(􀜤~~ ~~−~~2 ~~􀜥)]=\u003Cbr>􀜾 + 􀜿 􀝐􀜽􀝊 [(􀜤~~ ~~+~~2 ~~􀜥)]\u003Cbr>􀜽 − 􀜿   􀝐􀜽􀝊 [(􀜣~~ ~~−~~2 ~~􀜥)]=\u003Cbr>􀜽 + 􀜿 􀝐􀜽􀝊 [(􀜣~~ ~~+~~2 ~~􀜥)]\u003Cbr>􀝐􀜽􀝊 (􀜣) ∙ 􀝐􀜽􀝊(􀜤) ∙ 􀝐􀜽􀝊(􀜥) = 􀝐􀜽􀝊 (􀜣) + 􀝐􀜽􀝊(􀜤) + 􀝐􀜽􀝊(􀜥) |  |\n| Law of Cotangents | 􀜿􀝋􀝐 (􀜣~~2~~) = 􀜿􀝋􀝐 (􀜤~~2~~) = 􀜿􀝋􀝐 (􀜥~~2~~)\u003Cbr>􀝏 − 􀜽 􀝏 − 􀜾 􀝏 − 􀜿􀜿􀝋􀝐 (􀜣) ∙ 􀜿􀝋􀝐 (􀜤) ∙ 􀜿􀝋􀝐 (􀜥) = 􀜿􀝋􀝐 (􀜣) + 􀜿􀝋􀝐 (􀜤) + 􀜿􀝋􀝐 (􀜥) |  |\n\n\n| Pythagorean Theorem | If a right triangle, then\u003Cbr>􀢉􀫛 = 􀢇􀫛 + 􀢈􀫛\u003Cbr>Special Case: Same as Law of Cosines with angle C = 90° .\u003Cbr>􀜿 2 = 􀜽 2 + 􀜾 2 − 2􀜽􀜾 ∙ 􀜿􀝋􀝏 (90°)\u003Cbr>􀜿 2 = 􀜽 2 + 􀜾 2 − 2􀜽􀜾 ∙ 0 = 􀜽 2 + 􀜾 2 |  |  |\n| --- | --- | --- | --- |\n| Sum of Angles | 􀜣 ° + 􀜤 ° + 􀜥° = 180°􀜥° = 180° − (􀜣 ° + 􀜤 °) |  | 􀜣 + 􀜤 + 􀜥 = 􀟨 radians\u003Cbr>􀜥 = 􀟨 − (􀜣 + 􀜤) |\n| Area Formula | 􀜣 = 12 􀜾ℎ = 12 􀜽􀜾 ∙ 􀝏􀝅􀝊(􀜥) |  | |\n| Semi-Perimeter | 􀝏 = | 􀜽 + 􀜾 + 􀜿 2 |  |\n| Heron’s Formula | 􀜣 = √􀝏 (􀝏 − 􀜽)(􀝏 − 􀜾)(􀝏 − 􀜿) |  |  |\n| Mollweide’s Formula | 􀜽 + 􀜾   􀜿􀝋􀝏 [(􀜣~~ ~~−~~2 ~~􀜤)]\u003Cbr>=\u003Cbr>􀜿 􀝏􀝅􀝊 (􀜥~~2~~) |  |  |\n\nSolving for Angles, Sides, and Area  \n\n| Order of Use | Comments |\n| --- | --- |\n| 1. Sum of Angles | Easiest formula. |\n| 2. Pythagorean Theorem | Use if one of the angles is a right angle (90°) . |\n| 3. Law of Sines | Least complex. Use before the Law of Cosines, if possible. |\n| 4. Law of Cosines | More complex. Use only once, then use the Law of Sines. |\n| 5. Heron’s Formula | Use for area if all three sides are known. |\n| 6. Law of Tangents | Very complex and seldom used. |\n| 7. Law of Cotangents | Most complex and seldom used. |\n\n\n| Given | Find |  |  |\n| --- | --- | --- | --- |\n|  | Angle | Side | Area |\n| SSS | Law of Cosines | Given | Heron’s Formula |\n| SAS | NA | Law of Cosines | 􀜣 = 12 􀜽􀜾 ∙ 􀝏􀝅􀝊 (􀜥) |\n| SSA | Law of Sines | NA | 􀜣 = 12 􀜾ℎ |\n| ASS |  |  |  |\n| SAA | Sum of Angles | Law of Sines |  |\n| ASA |  |  |  |\n| AAS |  |  |  |\n| AAA | Given | Unsolvable. Not unique. One side needed. |  |\n\nAmbiguous Cases for SSA  \n\n| Scenario | \\# of Solutions | Illustrations |  |  |  |\n| --- | --- | --- | --- | --- | --- |\n| SSA | 0 – 2 Solutions | |  |  |  |\n|  |  | ℎ = 􀜽 sin 􀟠 |  |  |  |\n| 􀢈 \u003C 􀢎 | No Solution | |  |  |  |\n| 􀢈 = 􀢎 | One Solution | |  |  |  |\n| 􀜊 > 􀜐 | Two Solutions | |  |  |  |\n|  |  | 􀜾\u003Cbr>􀝏􀝅􀝊 (􀜤) | = | 􀜾\u003Cbr>􀝏􀝅􀝊 (180° − 􀜤) | |\n| a ≥ b | One Oblique\u003Cbr>Solution | |  |  |  |\n|  |  | When a = b, equilateral / isososocles\u003Cbr>When a > b, obtuse |  |  |  |\n\nFixed Angles Triangles  \n\n| 45-45-90 Triangle | 30-60-90 Triangle |\n| --- | --- |\n| | |\n| Proof: | Proof: |\n| a² + b² = c² | a² + b² = c² |\n| x = y | x = ½c |\n| x² + x² = 1² | y² + (½)² = 1² |\n| 2x² = 1 | y² + ¼ = 1 |\n| x² = ½ | y² = ¾ |\n|  |  |\n| √􀝔~~ ~~2 = √12 | √􀝕~~ ~~2 = √34 |\n| √2 | √3 |\n| 􀝔 =  2  | 􀝕 =  2  |\n\nFixed-Sided Triangles  \n\n| Pythagorean Triples |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| Definition:\u003Cbr>A Pythagorean triple is a right triangle with only integer sides.\u003Cbr>The examples on the right are expressed in the lowest form.\u003Cbr>You can scale any of these by an integer (e.g., 2,3,4,5) to generate a family of similar triangles.\u003Cbr>|  | 3 : 4 : 5\u003Cbr>5 : 12 : 13\u003Cbr>7 : 24 : 25\u003Cbr>8 : 15 : 17\u003Cbr>9 : 40 : 41\u003Cbr>11 : 60 : 61 12 : 35 : 37 13 : 84 : 85 15 : 112 : 113 16 : 63 : 65 17 : 144 : 145\u003Cbr>19 : 180 : 181 20 : 21 : 29 20 : 99 : 101\u003Cbr>21 : 220 : 221\u003Cbr>23 : 264 : 265 | 24 : 143 : 145 28 : 45 : 53 28 : 195 : 197\u003Cbr>32 : 255 : 257 33 : 56 : 65\u003Cbr>36 : 77 : 85 39 : 80 : 89 44 : 117 : 125 48 : 55 : 73 51 : 140 : 149\u003Cbr>52 : 165 : 173\u003Cbr>57 : 176 : 185\u003Cbr>60 : 91 : 109\u003Cbr>60 : 221 : 229 65 : 72 : 97 68 : 285 : 293 | 69 : 260 : 269 84 : 187 : 205 85 : 132 : 15","cbCailJTH45HZ6iL","https://ap.wps.com/l/cbCailJTH45HZ6iL","pdf",679103,6,"English","# Trig Laws and Formulas\n## Reference Triangle Laws\n## Pythagorean Theorem and Angle Sum\n## Area, Semi-Perimeter, and Heron’s Formula\n## Solving for Angles, Sides, and Area\n## Ambiguous Cases for SSA\n## Fixed Angles Triangles\n## Fixed-Sided Triangles\n## Interesting Trig Lengths on a Unit Circle\n## Radians and Arc Length","[{\"question\":\"Which formula should be used first when solving a triangle with angles and sides?\",\"answer\":\"Start with the Sum of Angles. Use the Pythagorean Theorem only for right triangles, then consider Law of Sines before Law of Cosines when possible, reserving Law of Cosines for cases where it is needed.\"},{\"question\":\"What do you do in an SSA ambiguous case?\",\"answer\":\"Determine the number of solutions based on relative side lengths (typically comparing a and b) and the height term h = a sin(θ). The document notes outcomes ranging from 0 to 2 solutions, including one solution when a = b.\"},{\"question\":\"How are radians related to arc length on a unit circle?\",\"answer\":\"The cheat sheet states that radians equal arc length s on a unit circle, with s = rθ and C = 2πr. It also provides conversions between degrees and radians using multiples of π.\"}]","Harold’s Triangles Cheat Sheet - Trig Laws and Formulas | PDF",1791927070]