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The cheat sheet summarizes conversions between degrees and radians, key special angles, and exact trigonometric values. It then covers formulas for arc length and sector/segment area, followed by guidance on solving trigonometric equations in radian mode. It also introduces small-angle approximations and demonstrates their use with several worked examples, including an exam-style problem involving prism cross-sections.","Radians Cheat Sheet  \nRadians are simply another way to measure angles. In many areas of mathematics and physics, using radians as opposed to degrees turns out to be much more convenient. For example, the arc length formula which we will cover soon is greatly simplified by using radians.  \nUsing Radians  \nTo convert between radians and degrees, you can use the fact that:  \n▪ 1 􀝎􀜽􀝀􀝅􀜽􀝊 = 18􀁓0°  \nIt helps to remember the following angles in radians:  \n▪  \n▪  \n▪  \n30° =  \n45° =  \n60° =  \n􀰗  \n6  \n􀁓  \n4  \n􀁓  \n3  \n▪ 90° = 􀰗2 ▪ 180° = 􀟨  \n▪ 360° = 2􀟨  \nYou can be asked to sketch trigonometric functions giving your angles in radians, so you should be very efficient at converting between radians and degrees.  \nYou also need to learn the exact value of certain trigonometric ratios given in radians:  \n▪ 􀝏􀝅􀝊 (􀰗6) = 12 ▪ 􀝏􀝅􀝊 (􀰗3) = 3  \n▪ 􀝏􀝅􀝊 (􀰗4) = 2  \n▪ 􀜿􀝋􀝏 (􀰗6) = 3  \n▪ 􀜿􀝋􀝏 (􀰗3) = 12 ▪ 􀜿􀝋􀝏 (􀰗4) = 2  \n▪ 􀝐􀜽􀝊 (􀰗6) = 3  \n▪ 􀝐􀜽􀝊 (􀰗3) = √3  \n▪ 􀝐􀜽􀝊 (􀰗4) = 1  \nArc length  \nTo find the arc length 􀝈 of a sector of a circle, we can use the formula  \n▪ 􀝈 = 􀝎􀟠  \nwhere 􀝎 is the radius of the circle and 􀟠 is the angle contained in the sector, given in radians.  \nAreas  \nTo find the area of a sector, shown in blue, we can use the formula  \n▪ 􀜣 = 12 􀝎 2 􀟠  \nTo find the area of a segment, shown in yellow, we can use the formula  \n▪ 􀜣 = 12 􀝎 2 (􀟠 − 􀝏􀝅􀝊􀟠)  \nrecall that the area of a triangle is 12 􀜽􀜾􀝏􀝅􀝊􀟠  \nYou need to be able to apply the above formulae to problems.  \nExample 1: In the diagram below, AB is the diameter of a circle centre O of radius r cm and ∠􀜤􀜱􀜥 = 􀟠 radians. Given that the area of △ 􀜥􀜱􀜤 is equal to that of the shaded segment, show that  \n􀟠 + 2􀝏􀝅􀝊􀟠 = 􀁓.  \nArea 􀜥􀜱􀜤 = 12 􀝎 2􀝏􀝅􀝊􀟠 (area of a triangle, 􀜱􀜥 = 􀜱􀜤 = 􀝎)  \nShaded segment = 12 􀝎 2 ((􀟨 − 􀟠) − sin (􀟨 − 􀟠))  \nEquating: 12 􀝎 2􀝏􀝅􀝊􀟠 = 12 􀝎 2 ((􀟨 − 􀟠) − sin(􀟨 − 􀟠)) Notice that sin(􀟨 − 􀟠) = sin(􀟠),  \ndividing through by 12 􀝎 2 and rearranging: 􀝏􀝅􀝊􀟠 = 􀟨 − 􀟠 − 􀝏􀝅􀝊􀟠  \n∴ 􀟠 + 2􀝏􀝅􀝊􀟠 = 􀟨 as required  \nSolving trigonometric equations  \nYou also need to be able to solve trigonometric equations using radians. The method is exactly the same as with degrees, but you need to give your answers in radians.  \n▪ If the interval is given in radians, then you should leave your answers in radians.  \nLet’s go through an example.  \nExample 2: Solve 8􀝐􀜽􀝊2􀝔 = 7 in the interval 0 ≤ 􀝔 ≤ 2􀁓.  \nThe interval is given in radians, so we must make sure we work in radians. Don’t forget to switch to  \nthe radians mode on your calculator.􀝐􀜽􀝊2􀝔 = 78  \n2􀝔 = arctan (78) = 0.719  \nUsing CAST or a graphical method, our solutions are:  \n2􀝔 = 0.719, 􀁓 + 0.719 ⇒ 2􀝔 = 0.719, 3.86  \n∴ 􀝔 = 0.36, 1.93 are our final solutions.  \n\n| S | A\u003Cbr> 0.719 |\n| --- | --- |\n| 0.719 \u003Cbr>T | C |\n\nAs you can see, the procedure is still the same. You just need to be comfortable with using radians.  \nSmall angle approximations  \nWhen 􀟠 is close to zero and measured in radians, we can use the following approximations:  \n▪ 􀝏􀝅􀝊􀟠 ≈ 􀟠  \n▪ 􀜿􀝋􀝏􀟠 ≈ 1 − 􀰏22  \n▪ 􀝐􀜽􀝊􀟠 ≈ 􀟠  \nTo see why these are good approximations, we can graph each of the functions against their approximations.  \n􀝕 = 􀜿􀝋􀝏􀝔  \n􀝕 = 1 − 􀝔22  \nWe can see that around 􀝔 = 0, the graphs are almost identical. This explains why these approximations are suitable for 􀝔 close to zero. There is no set range for which these approximations are to be used.  \nExample 3: When 􀟠 is close to zero, show that 4􀯖􀯢􀯦312􀯡+25􀰏􀯦􀯜􀯡􀰏 can be rewritten as 9􀟠 + 2 .  \nInputting our approximations:  \n⇒ 4􀜿􀝋􀝏3􀟠 − 2 + 5􀝏􀝅􀝊􀟠 ≈ 4 (1 − (3􀰏2)2) − 2 + 5􀟠 = 2 + 5􀟠 − 18􀟠2  \n⇒ 1 − 􀝏􀝅􀝊2􀟠 ≈ 1 − 2􀟠  \n∴ 4􀯖􀯢􀯦312􀯡+25􀰏􀯦􀯜􀯡􀰏 ≈ 2+51􀰏−1􀰏8􀰏2 = (1−2􀰏1−)(22􀰏+9􀰏) = 2 + 9􀟠 as required.  \nOnce you substitute your approximations, it becomes a matter of simplifying your expression to achieve the desired result.  \nExample 4: When 􀟠 is close to zero, find the approximate value of 􀜿􀝋􀝏4 􀟠 − 􀝏􀝅􀝊4 􀟠 .  \n⇒ 􀜿􀝋􀝏4 􀟠 − 􀝏􀝅􀝊4 􀟠 = (􀜿􀝋􀝏2 􀟠 + 􀝏􀝅􀝊2 􀟠)(􀜿􀝋􀝏 2 􀟠 − 􀝏􀝅􀝊2 􀟠)  \n= (1)(􀜿􀝋􀝏2 􀟠 − 􀝏􀝅􀝊2 􀟠)  \n⇒ [1 − 􀝏􀝅􀝊2 􀟠 − 􀝏􀝅􀝊2 􀟠] = 1 − 2􀝏􀝅􀝊2 􀟠 = 1 − 2􀟠 2  \ndifference of two squares using 􀜿􀝋􀝏2 􀟠 ","cbCaidDWbWbNSB3y","https://ap.wps.com/l/cbCaidDWbWbNSB3y","pdf",452167,3,"English","en",105,"# Using Radians\n## Convert between radians and degrees\n## Recall key radian angles\n# Arc length and areas\n## Arc length formula\n## Sector area formula\n## Segment area formula\n# Solving with radians\n## Trigonometric equations in radians\n## Small angle approximations\n# Worked examples\n## Example questions and exam-style practice","[{\"question\":\"How do you convert between degrees and radians?\",\"answer\":\"Use the conversion relationship given in the cheat sheet: 1 rad = 180°/π degrees. The document also recommends efficient practice converting special angles.\"},{\"question\":\"What formula is used to find arc length and sector area?\",\"answer\":\"Arc length is given by s = rθ, where r is the radius and θ is in radians. Sector area is given by A = (1/2)r^2θ.\"},{\"question\":\"What small-angle approximations are provided for angles near zero?\",\"answer\":\"For θ close to 0 (in radians), the sheet uses sinθ ≈ θ, cosθ ≈ 1 − θ^2/2, and tanθ ≈ θ. It explains these by comparing graphs near θ = 0.\"}]","Radians Cheat Sheet - Trigonometry Angles Conversion | PDF",1791938246,{"code":4,"msg":75,"data":76},"ok",{"site_id":69,"language":68,"slug":77,"title":59,"keywords":78,"description":60,"schema_data":79,"social_meta":134,"head_meta":136,"extra_data":138,"updated_unix":139},"radians-cheat-sheet-trigonometry-angles-conversion","",{"@graph":80,"@context":133},[81,96,116],{"@type":82,"itemListElement":83},"BreadcrumbList",[84,88,91,93],{"item":85,"name":86,"@type":87,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":89,"name":10,"@type":87,"position":90},"https://docshare.wps.com/template/",2,{"item":92,"name":11,"@type":87,"position":66},"https://docshare.wps.com/template/presentations/",{"item":94,"name":59,"@type":87,"position":95},"https://docshare.wps.com/template/radians-cheat-sheet-trigonometry-angles-conversion/573367/",4,{"url":94,"name":59,"@type":97,"image":98,"author":103,"headline":59,"publisher":105,"fileFormat":108,"inLanguage":68,"description":60,"dateModified":109,"datePublished":110,"encodingFormat":108,"isAccessibleForFree":111,"interactionStatistic":112},"DigitalDocument",{"url":99,"@type":100,"width":101,"height":102},"https://docshare.wps.com/thumbnails/radians-cheat-sheet-trigonometry-angles-conversion/573367.png","ImageObject",442,249,{"name":57,"@type":104},"Person",{"url":85,"name":106,"@type":107},"DocShare","Organization","application/pdf","2026-10-18","2026-10-14",true,{"@type":113,"interactionType":114,"userInteractionCount":66},"InteractionCounter",{"@type":115},"ViewAction",{"@type":117,"mainEntity":118},"FAQPage",[119,125,129],{"name":120,"@type":121,"acceptedAnswer":122},"How do you convert between degrees and radians?","Question",{"text":123,"@type":124},"Use the conversion relationship given in the cheat sheet: 1 rad = 180°/π degrees. The document also recommends efficient practice converting special angles.","Answer",{"name":126,"@type":121,"acceptedAnswer":127},"What formula is used to find arc length and sector area?",{"text":128,"@type":124},"Arc length is given by s = rθ, where r is the radius and θ is in radians. Sector area is given by A = (1/2)r^2θ.",{"name":130,"@type":121,"acceptedAnswer":131},"What small-angle approximations are provided for angles near zero?",{"text":132,"@type":124},"For θ close to 0 (in radians), the sheet uses sinθ ≈ θ, cosθ ≈ 1 − θ^2/2, and tanθ ≈ θ. It explains these by comparing graphs near θ = 0.","https://schema.org",{"og:url":94,"og:type":135,"og:title":59,"og:site_name":106,"og:description":60},"article",{"robots":137,"canonical":94},"index,follow",{"doc_id":55,"site_id":69},1792022901]