Section 4.2 Radians, Arc Length, and Area of a Sector
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1 Sectin 4.2 Radian, Ac Length, and Aea f a Sect An angle i fmed by tw ay that have a cmmn endpint (vetex). One ay i the initial ide and the the i the teminal ide. We typically will daw angle in the cdinate plane with the initial ide alng the pitive x axi. Teminal ide A Vetex B Initial Side C We may name thi angle any ne f the fllwing way: B ABC CBA When uing the ntatin ABC and CBA, the vetex i alway the middle lette. Meauing Angle We meaue angle in tw diffeent way, bth f which ely n the idea f a cmplete evlutin in a cicle. The fit i degee meaue. In thi ytem f angle meaue ne cmplete 1 evlutin i 360. S ne degee i f the cicle. 360 The ecnd methd i called adian meaue. One cmplete evlutin i 2. The pblem in thi ectin ae wked in adian. Radian i a unit fee meauement. Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 1
2 The Radian Meaue f an Angle Place the vetex f the angle at the cente f a cicle (cental angle) f adiu. Let dente the length f the ac intecepted by the angle. The adian meaue f the angle i the ati f the ac length t the adiu. In ymbl,. In thi definitin it i aumed that and have the ame linea unit. 12 Example 1: A cental angle, =, in a cicle intecept an ac f length m. What i the 2 5 adiu f the cicle? Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 2
3 Relatinhip between Degee and Radian Hw can we btain a elatinhip between degee and adian? We cmpae the numbe f degee and the numbe f adian in ne cmplete tatin in a cicle. We knw that360 i all the way aund a cicle. The length f the intecepted ac i equal t the cicumfeence f the cicle. Theefe, the adian meaue f thi cental angle i the cicumfeence f the cicle divided by the cicle adiu,. The cicumfeence f a cicle f a adiu i 2. We ue the fmula f adian meaue t find the adian meaue f the360 angle. S, 360 = 2 adian. the cicle' cicumfeence 2 2 Dividing bth ide by 2, we get 180 = adian. Dividing thi lat equatin by180 give the cnvein ule that fllw. Cnvein between Degee and Radian Uing the fact that adian = 180, adian 1. T cnvet degee t adian, multiply degee by T cnvet adian t degee, multiply adian by. adian Nte: The unit yu ae cnveting t appea in the numeat f the cnvein fact. Example 2: Cnvet -135 t adian. Example 3: Cnvet 7 t degee. 9 Cmmn Angle (Memize thee!) 360 = = Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 3
4 Sect f a Cicle A ect f a cicle i the pat f a cicle encled by tw adii f a cicle and thei intecepted ac. Example 4: Find the peimete f a ect with cental angle 60 and adiu 3 m. Recall: Aea f a Sect f a Cicle Fmula In a cicle f adiu, the aea A f a ect with cental angle f adian meaue i given by 1 A 2. 2 Example 5: Given a cicle the aea f ect i 3 in 2 and the cental angle i 30. Find the adiu Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 4
5 Linea and Angula Velcity (Speed) Cnide a mey-g-und edupic.cm The ide tavel in a cicula mtin. Sme f the he ae ight alng the edge f the meyg-und, and me ae cle t the cente. If yu ae n ne f the he at the edge, yu will tavel fathe than mene wh i n a he nea the cente. But the length f time that bth peple will be n the ide i the ame. If yu wee n the edge, nt nly did yu tavel fathe, yu al taveled fate. Hweve, eveyne n the mey-g-und tavel thugh the ame numbe f degee ( adian). Thee ae tw quantitie we can meaue fm thi, angula velcity and linea velcity. The angula velcity f a pint n a tating bject i the numbe f degee ( adian evlutin) pe unit f time thugh with the pint tun. Thi will be the ame f all pint n the tating bject. We let the Geek lette (mega) epeent angula velcity. Uing the definitin abve,. Unit f angula velcity will be f the fm: km/h, m/, mph, etc. t Al, evlutin. The linea velcity f a pint n the tating bject i the ditance pe unit f time that the pint tavel alng it cicula path. Thi ditance will depend n hw fa the pint i fm the axi f tatin (f example, the cente f the mey-g-und). We dente linea velcity by v. Uing the definitin abve, v, whee i the ac length (.) t We can etablih a elatinhip between the tw kind f peed by ubtituting f : t t int v t Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 5
6 Example 6: A Fei wheel make 3 evlutin pe minute. The paenge it in eat that ae 25 feet fm the cente f the wheel. a. Find the angula peed in unit f adian/minute. b. What i the linea velcity f the paenge in the eat? Example 7: A paticle i mving n the peimete f a cicle with adiu 6 and angula peed 3 adian pe ecnd. Afte cmpleting 3 full tatin, the paticle taveled f 7 me ecnd and tpped. What i the length f the ttal ditance the paticle taveled? Sectin 4.2 Radian, Ac Length, and the Aea f a Sect 6
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