In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Transcription

1 Radians mc-ty-adians Atschoolweusuallyleantomeasueanangleindegees. Howeve,theeaeothewaysof measuinganangle. Onethatweaegoingtohavealookatheeismeasuinganglesinunits called adians. In many scientific and engineeing calculations adians ae used in pefeence to degees. In ode to maste the techniques explained hee it is vital that you undetake plenty of pactice execises so that they become second natue. Afteeadingthistext,and/oviewingthevideotutoialonthistopic,youshouldbeableto: use adians to measue angles convetanglesinadianstoanglesindegeesandvicevesa findthelengthofanacofacicle findtheaeaofasectoofacicle findtheaeaofasegmentofacicle Contents 1. Intoduction 2 2. Definition of a adian 2 3. Ac length 3 4. Equivalent angles in degees and adians 4 5. Findinganaclengthwhentheangleisgivenindegees 5 6. Theaeaofasectoofacicle 6 7. Miscellaneous examples c mathcente 2009

2 1. Intoduction Atschoolweusuallyleantomeasueanangleindegees.Weaewellawaethatafullotation is 360 asshowninfigue o Figue1.Afullotationis 360. Howeve,theeaeothewaysofmeasuinganangle.Onewaythatweaegoingtohavealook at hee is measuing angles in units called adians. In many scientific and engineeing calculations adians ae used in pefeence to degees. 2. Definition of a adian Consideacicleofadius asshowninfigue2. 1 ad Figue2.Theacshownhasalengthchosentoequaltheadius;theangleisthen1adian. InFigue2wehavehighlightedpatofthecicumfeenceoftheciclechosentohavethesame lengthastheadius.theangleatthecente,sofomed,is1adian. Key Point Anangleofoneadianissubtendedbyanachaving thesamelengthastheadiusasshowninfigue c mathcente 2009

3 3. Ac length Wewillnowusethisdefinitiontofindafomulafothelengthofanabitayac. Wehaveseenthatanangleof1adianissubtendedbyanacoflength asillustatedinthe left-mostdiagaminfigue3.byextensionanangleof2adianswillbesubtendedbyanacof length 2, as shown Figue3.Anangleof2adiansissubtendedbyanacoflength 2. Notefomthesediagamsthatthelengthoftheacisalwaysgivenby the angle in adians the adius Inthegenealcase,thelength s,ofanabitayacwhichsubtendsanangle is asillustated infigue4. s Figue4.Theaclength s,isgivenby Thisgivesusawayofcalculatingtheaclengthwhenweknowtheangleatthecenteofthe cicleandweknowitsadius. Key Point aclength s = (note: must be measued in adians) Execise 1 Deteminetheangle(inadians)subtendedatthecenteofacicleofadius3cmbyeachof the following acs: a) acoflength6cm b) acoflength 3cm c) acoflength1.5cm d) acoflength 6cm 3 c mathcente 2009

4 4. Equivalent angles in degees and in adians Weknowthattheaclengthfoafullcicleisthesameasitscicumfeence, 2. Wealsoknowthattheaclength=. Sofoafullcicle that is 2 = = 2 Inothewods,whenweaewokinginadians,theangleinafullcicleis 2adians,inothe wods 360 = 2 adians This enables us to have a set of equivalences between degees and adians. Key Point fom which it follows that 360 = 2adians 180 = adians 90 = 2 adians 45 = 4 adians 60 = 3 adians 30 = 6 adians TheKeyPointgivesalistofanglesmeasuedindegeesontheleftandtheequivalentlistin adiansontheight. Itisimpotantinmathematicalwokthatyouecodcoectlytheunitof measue you ae using. Anothe useful elationship is given as follows: adians = 180 so 1adian = 180 degees = (3d.p.) So1adianisjustove 57. Some notation. Thee ae vaious conventions used to denote adians. Some books and some teaches use ads asin2ads. Othesuseasmall casin 2 c. Someothesusenosymbolatallandassumethat adiansaebeingused. Whenanangleisexpessedasamultipleof,foexampleasinthe expession sin 3 2,itistakenaseadthattheangleisbeingmeasuedinadians. 4 c mathcente 2009

5 Execise 2 1.Wheneachofthefollowinganglesisconvetedfomdegeestoadianstheanswecanbe expessedasamultipleof (notethatitmaybeafactionalmultiple).ineachcasestate themultiple(e.gfoanansweof 4themultipleis 4). 5 5 a) 90 o b) 360 o c) 60 o d) 45 o e) 120 o f) 15 o g) 135 o h) 270 o 2. Convet each of the following angles fom adians to degees. a) adians b) 3 adians c) adians d) adians e) 5adians f) adians g) 7 adians h) adians Conveteachofthefollowingangles fomdegees toadiansgivingyouansweto2 decimal places. a) 17 o b) 49 o c) 124 o d) 200 o 4.Conveteachofthefollowinganglesfomadianstodegees, givingyouansweto1 decimal place. a) 0.6adians b) 2.1adians c) 3.14adians d) 1adian 5. Finding an ac length when the angle is given in degees Weknowthatif ismeasuedinadians,thenthelengthofanacisgivenby s =. Suppose ismeasuedindegees.weshalldeiveanewfomulafotheaclength. o s Figue5.Inthiscicletheangle ismeasuedindegees. RefeingtoFigue5,theatiooftheaclengthtothefullcicumfeencewillbethesameas theatiooftheanglesubtendedbytheac,totheangleinafullcicle;thatis s 2 = 360 So,when ismeasuedindegeeswecanusethefollowingfomulafoaclength: s = Notice how the ealie fomula, used when the angle is measued in adians, is much simple. 5 c mathcente 2009

6 6. The aea of a secto of a cicle Asectoofaciclewithangle isshownshadedinfigue6. Figue6.Theshadedaeaisasectoofthecicle. Theatiooftheaeaofthesectototheaeaofthefullciclewillbethesameastheatioof theangle totheangleinafullcicle.thefullciclehasaea 2.Theefoe andso aea of secto aeaoffullcicle = 2 aea of secto = 2 2 = Key Point aea of secto = when ismeasuedinadians 7. Miscellaneous Examples Example ConsidethecicleshowninFigue7.Supposewewishtocalculatetheangle Figue 7. Calculate the angle. 6 c mathcente 2009

7 Weknowtheaclengthandadius. Wecanusethefomula s =. Substitutingthegiven values 25 = 10 andso = = 2.5ads Whatisthisangleindegees?Weknow andso It follows that Example ads = 180 1ad = ads = = RefetoFigue8.Supposewehaveacicleofadius10cmandanacoflength15cm.Suppose wewanttofind(a)theangle,(b)theaeaofthesecto OAB,(c)theaeaofthemino segment(shaded). O 10 B 10 A 15 Figue8.Theshadedaeaiscalledtheminosegment. (a)using s = wehave 15 = 10andso = = 1.5c. (b)usingthefomulafotheaeaofthesecto, A = 1 2 2,wefind aea = = 1 2 (102 )(1.5) = 75cm 2 (c)wealeadyknowthattheaeaofthesecto OABis 75cm 2.Ifwecanwokouttheaeaof thetiangle AOBwecanthendeteminetheaeaoftheminosegment. (Recallthefomulae fotheaeaoftiangle, A = 1 ab sin C.) 2 aeaoftiangle = sin = sin 1.5 = cm c mathcente 2009

8 Theefoetheaeaoftheminosegmentis Example = cm 2 (to2dp.) Supposewehaveanangleof 120.Whatisthisangleinadians?Weknowthat andso then ads = 180 ads = = 120 ads 180 Thiscanbewittenas 2 3 adians(=2.094adians). Execise 3 Asectoofacicleisanaeaboundedbytwoadiiandanac. Asectohasanangleatthe centeofthecicle.allthequestionsbelowelatetoaciclewithadius5cm. 1.Deteminethelengthoftheac(coectto2decimalplaces)whentheangleatthecente isa)1.2adians b) adians c)45o 2 2.Calculatetheaea(coectto2decimalplaces)ofeachofthetheesectosinQuestion 1. 3.Asectoofthisciclehasaea50cm 2.Whatistheangle(inadians)atthecenteofthis secto? Answes Execise 1 a)2 b) c)0.5 d) 2 Execise 2 1.a) 1 b)2 c) 1 d) 1 e) 2 f) 1 g) 3 h) a)90 o b)135 o c)180 o d)30 o e)900 o f)144 o g)315 o h)18 o 3.a)0.30adians b)0.86adians c)2.16adians d)3.49adians 4.a)15.3 o b)120.3 o c)179.9 o d)57.3 o Execise 3 1.a)6cm b)7.85cm c)3.93cm 2.a)15cm 2 b)19.63cm 2 c)9.82cm 2 3.4adians 8 c mathcente 2009