SECTION 5-3 Angles and Their Measure

Transcription

1 5-3 Angle and Thei Meaue 357 APPLICATIONS Appoximating. Poblem 93 and 9 efe to a equence of numbe geneated a follow: If an n-d egula polygon i incibed in a cicle of adiu, then it can be hown that the aea of the polygon i given by A 2 n2 in 2 n co a n a n a a 2 a co a a 3 a 2 co a 2. a n a n co a n Compute each aea exactly and then to ignificant digit uing a calculato if the aea i not an intege. 89. n 2, 5 mete 90. n, 3 inche 9. n 3, inche 92. n 8, 0 centimete Let a 0.5, and compute the fit five tem of the equence to ix decimal place and compae the fifth tem with /2 computed to ix decimal place. 9. Repeat Poblem 93, tating with a. SECTION 5-3 Angle and Thei Meaue Angle Degee and Radian Meaue Fom Degee to Radian, and Vice Vea In thi ection, we will intoduce the idea of angle and two meaue of angle, degee and adian. Angle The tudy of tigonomety depend on the concept of angle. An angle i fomed by otating (in a plane) a ay m, called the initial of the angle, aound it endpoint until it coincide with a ay n, called the teminal of the angle. The common endpoint V of m and n i called the vetex (ee Fig. ). A counteclockwie otation poduce a poitive angle, and a clockwie otation poduce a negative angle, a hown in Figue 2(a) and (b). The amount of otation in eithe diection i not eticted. FIGURE Angle o angle PVQ o V. Teminal Q n V Initial P m

2 358 5 Tigonometic Function FIGURE 2 Angle and otation. Teminal Initial Teminal Initial Teminal negative (a) (b) (c) Two diffeent angle may have the ame initial and teminal, a hown in Figue 2(c). Such angle ae aid to be coteminal. An angle in a ectangula coodinate ytem i aid to be in tandad poition if it vetex i at the oigin and the initial i along the poitive x axi. If the teminal of an angle in tandad poition lie along a coodinate axi, the angle i aid to be a quadantal angle. If the teminal doe not lie along a coodinate axi, then the angle i often efeed to in tem of the quadant in which the teminal lie (Fig. 3). Initial FIGURE 3 Angle in tandad poition. Degee and Radian Meaue II III Teminal y Initial I IV x II III Teminal y Initial i a quadantal angle i a thid-quadant angle i a econd-quadant angle (a) (b) (c) Jut a line egment ae meaued in centimete, mete, inche, o mile, angle ae meaued in diffeent unit. The two mot commonly ued unit fo angle meaue ae degee and adian. I IV x II III Teminal y Initial I IV x DEFINITION Degee Meaue An angle fomed by one complete otation i aid to have a meaue of 360 degee (360 ). An angle fomed by 360 of a complete otation i aid to have a meaue of degee ( ). The ymbol denote degee. Cetain angle have pecial name. Figue how a taight angle, a ight angle, an acute angle, and an obtue angle. FIGURE Type of angle Staight angle Right angle Acute angle Obtue angle otation otation (0 90) (90 80) 2 (a) (b) (c) (d)

3 5-3 Angle and Thei Meaue 359 Two poitive angle ae complementay if thei um i 90 ; they ae upplementay if thei um i 80. A degee can be divided futhe uing decimal notation. Fo example, 2.75 epeent an angle of degee meaue 2 plu thee-quate of degee. A degee can alo be divided futhe uing minute and econd jut a an hou i divided into minute and econd. Each degee i divided into 60 equal pat called minute, and each minute i divided into 60 equal pat called econd. Symbolically, minute ae epeented by and econd by. Thu, 2 23 i a concie way of witing 2 degee, 23 minute, and econd. Decimal degee (DD) ae ueful in ome intance and degee minute econd (DMS) ae ueful in othe. You hould be able to go fom one fom to the othe, a demontated in Example. Conveion Accuacy If an angle i meaued to the neaet econd, the conveted decimal fom hould not go beyond thee decimal place, and vice vea. EXAMPLE Fom DMS to DD and Back (A) Convet 2 72 to decimal degee. (B) Convet to degee minute econd fom. 7 Solution (A) (B) (0.8360) (0.9860) Matched Poblem (A) Convet to DD fom. (B) Convet to DMS fom. Some cientific and ome gaphing calculato can convet the DD and DMS fom automatically, but the poce diffe ignificantly among the vaiou type of calculato. Check you owne manual fo you paticula calculato. The conveion method outlined in Example how you the eaoning behind the poce and ae ometime eaie to ue than the automatic method fo ome calculato.

4 360 5 Tigonometic Function Degee meaue of angle i ued extenively in engineeing, uveying, and navigation. Anothe unit of angle meaue, called the adian, i bette uited fo cetain mathematical development, cientific wok, and engineeing application. DEFINITION 2 Radian Meaue If the vetex of an angle i placed at the cente of a cicle with adiu 0, and the length of the ac oppoite on the cicumfeence i, then the adian meaue of i given by Alo, adian O If, then adian O adian Thu, one adian i the ize of the cental angle of a cicle that intecept an ac the ame length a the adiu of the cicle. [Note: and mut be meaued in the ame unit. Alo note that i being ued in two way: a the name of an angle and a the meaue of the angle. The context detemine the choice. Thu, when we wite /, we mean the adian meaue of angle i /.] EXAMPLE 2 Computing Radian Meaue What i the adian meaue of a cental angle oppoite an ac of 2 mete in a cicle of adiu 6 mete? Solution 2 mete 6 mete adian

5 5-3 Angle and Thei Meaue 36 Matched Poblem 2 What i the adian meaue of a cental angle oppoite an ac of 60 feet in a cicle of adiu 2 feet? Remak. Radian meaue i a unitle numbe. The unit in which the ac length and adiu ae meaued cancel; hence, we ae left with a unitle, o pue, numbe. Fo thi eaon, the wod adian i often omitted when we ae dealing with the adian meaue of angle unle a pecial emphai i deied. EXPLORE-DISCUSS Dicu why the adian meaue of an angle i independent of the ize of the cicle having the angle a a cental angle. Fom Degee to Radian, and Vice Vea What i the adian meaue of an angle of 80? A cental angle of 80 i ubtended by an ac one-half the cicumfeence of a cicle. Thu, if C i the cicumfeence of a cicle, then one-half the cicumfeence i given by C and ad Hence, 80 coepond to * ad. Thi i impotant to emembe, ince the adian meaue of many pecial angle can be obtained fom thi coepondence. Fo example, 90 i 80 /2; theefoe, 90 coepond to /2 ad. EXPLORE-DISCUSS 2 Complete Table : TABLE Radian /6 / /3 /2 3/2 2 Degee *The contant ha a long and inteeting hitoy; a few impotant date ae lited below: 650 B.C. Rhind Papyu B.C. Achimede 3 0 < < ( < < ) A.D. 26 Liu Hui 3.59 A.D. 70 Tu Ch ung-chih A.D. 67 Leibniz (...) (Thi and othe eie can be ued to compute to any decimal accuacy deied.) A.D. 76 Johann Lambet Showed to be iational ( a a decimal i nonepeating and nonteminating)

6 362 5 Tigonometic Function The eult in Table ae ummaized in Figue 5 fo eay efeence. Thee coepondence and multiple of them will be ued extenively in wok that follow. FIGURE 5 Radian degee coepondence /2 60 /3 5 / 30 / /2 In geneal, the following popotion can be ued to convet degee meaue to adian meaue, and vice vea. Radian Degee Conveion Fomula deg 80 ad ad deg 80 ad ad ad ad 80 deg Baic popotion Radian to degee Degee to adian [Note: The baic popotion i uually eaie to emembe. Alo we will omit unit in calculation until the final anwe. If you calculato doe not have a key labeled, ue 3.59.] Some cientific and gaphing calculato can automatically convet adian meaue to degee meaue, and vice vea. Check you owne manual fo you paticula calculato. EXAMPLE 3 Radian Degee Conveion (A) Find the adian meaue, exact and to 3 ignificant digit, of an angle of 75. (B) Find the degee meaue, exact and to ignificant digit, of an angle of 5 adian. (C) Find the adian meaue to two decimal place of an angle of 2. Solution exact 3 ig. dig. (A) ad ad 80 deg 5 (75) exact ig. dig. (B) deg 80 ad ad (5) 286.5

7 5-3 Angle and Thei Meaue 363 FIGURE 6 Automatic conveion. (C) Change 2 to DD fit. ad ad To 2 dec. place 80 deg (.2) Figue 6 how the thee conveion above done automatically on a gaphing calculato. Matched Poblem 3 (A) Find the adian meaue, exact and to 3 ignificant digit, of an angle of 20. (B) Find the degee meaue, exact and to 3 ignificant digit, of an angle of adian. (C) Find the adian meaue to thee decimal place of an angle of Remak. We will wite in place of deg and ad when it i clea fom the context whethe we ae dealing with degee o adian meaue. EXAMPLE Engineeing A belt connect a pulley of 2-inch adiu with a pulley of 5-inch adiu. If the lage pulley tun though 0 adian, though how many adian will the malle pulley tun? Solution We fit ketch Figue 7. FIGURE 7 P 5 in. Q 2 in. When the lage pulley tun though 0 adian, the point P on it cicumfeence will tavel the ame ditance (ac length) that point Q on the malle cicle tavel. Fo the lage pulley: Fo the malle pulley: (5)(0) 50 inche adian

8 36 5 Tigonometic Function Matched Poblem In Example, though how many adian will the lage pulley tun if the malle pulley tun though adian? Anwe to Matched Poblem. (A) (B) ad (A).9 (B) 57.3 (C) ad 3 EXERCISE 5-3 In all poblem, if angle meaue i expeed by a numbe that i not in degee, it i aumed to be in adian. A Find the degee meaue of each of the angle in Poblem, keeping in mind that an angle of one complete otation coepond to otation 2. 5 otation otation. otation Find the adian meaue of a cental angle oppoite an ac in a cicle of adiu, whee and ae a given in Poblem centimete, 2 centimete 6. 8 inche, 6 inche 7. 2 feet, 30 feet 8. 8 mete, 27 mete Find the adian meaue of each angle in Poblem 9 2, keeping in mind that an angle of one complete otation coepond to 2 adian otation 0. 6 otation 3. otation 2. otation B Find the exact adian meaue, in tem of, of each angle in Poblem , 60, 90, 20, 50, , 20, 80, 20, 300, , 90, 35, , 80, 270, 360 Find the exact degee meaue of each angle in Poblem , 2 3,, 3, 5 3, ,, 3 2, 2, 2, 3, In Poblem 2 26, detemine whethe the tatement i tue o fale. If tue, explain why. If fale, give a counteexample. 2. If two poitive angle ae upplementay, then one i obtue and the othe i acute. 22. If two poitive angle ae complementay, then both ae acute. 23. If two angle in tandad poition ae coteminal, then they have the ame meaue. 2. If two angle in tandad poition have the ame meaue, then they ae coteminal. 25. If an angle i a ight angle, then it i a quadantal angle. 26. If an angle i a quadantal angle, then it i a ight angle. Convet each angle in Poblem to decimal degee to thee decimal place Convet each angle in Poblem 3 3 to degee minute econd fom , 3, 2, 2 3, 5 6,

9 5-3 Angle and Thei Meaue 365 Find the adian meaue to thee decimal place fo each angle in Poblem Find the degee meaue to two decimal place fo each angle in Poblem Indicate whethe each angle in Poblem 3 5 i a I, II, III, o IV quadant angle o a quadantal angle. All angle ae in tandad poition in a ectangula coodinate ytem. (A ketch may be of help in ome poblem.) C Which angle in Poblem ae coteminal with 20 if all angle ae placed in tandad poition in a ectangula coodinate ytem? Which angle in Poblem 6 66 ae coteminal with 3 if all angle ae placed in tandad poition in a ectangula coodinate ytem? APPLICATIONS 67. Cicumfeence of the Eath. The ealy Geek ued the popotion /C /360, whee i an ac length on a cicle, i the degee meaue of the coeponding cental angle, and C i the cicumfeence of the cicle (C 2). Eatothene (20 B.C.), in hi famou calculation of the cicumfeence of the Eath, eaoned a follow: He knew that at Syene (now Awan) duing the umme oltice the noon un wa diectly ovehead and hined on the wate taight down a deep well. On the ame day at the ame 8 3 time, 5,000 tadia (appox. 500 mile) due noth in Alexandia, un ay coed a vetical pole at an angle of 7.5, a indicated in the figue. Cay out Eatothene calculation fo the cicumfeence of the Eath to the neaet thouand mile. (The cuent calculation fo the equatoial cicumfeence i 2,902 mile). Eath Alexandia 7.5 Syene Well Sun ay 68. Cicumfeence of the Eath. Repeat Poblem 67 with the un coing the vetical pole in Alexandia at Cicumfeence of the Eath. In Poblem 67, vebally explain how in the figue wa detemined. 70. Cicumfeence of the Eath. Vebally explain how the adiu, uface aea, and volume of the Eath can be detemined fom the eult of Poblem Radian Meaue. What i the adian meaue of the lage angle made by the hand of a clock at :30? Expe the anwe exactly in tem of. 72. Radian Meaue. What i the adian meaue of the malle angle made by the hand of a clock at :30? Expe the anwe exactly in tem of. 73. Engineeing. Though how many adian doe a pulley of 0-centimete diamete tun when 0 mete of ope i pulled though it without lippage? 7. Engineeing. Though how many adian doe a pulley of 6-inch diamete tun when feet of ope i pulled though it without lippage? 75. Atonomy. A line fom the un to the Eath weep out an angle of how many adian in week? Aume the Eath obit i cicula and thee ae 52 week in a yea. Expe the anwe in tem of and a a decimal to two decimal place. 76. Atonomy. A line fom the cente of the Eath to the equato weep out an angle of how many adian in 9 hou? Expe the anwe in tem of and a a decimal to two decimal place. 77. Engineeing. A tail bike ha a font wheel with a diamete of 0 centimete and a back wheel of diamete 60 centimete. Though what angle in adian doe the font wheel tun if the back wheel tun though 8 adian?

10 366 5 Tigonometic Function 78. Engineeing. In Poblem 77, though what angle in adian will the back wheel tun if the font wheel tun though 5 adian? The ac length on a cicle i eay to compute if the coeponding cental angle i given in adian and the adiu of the cicle i known ( ). If the adiu of a cicle i lage and a cental angle i mall, then an ac length i often ued to appoximate the length of the coeponding chod, a hown in the figue. If an angle i given in degee meaue, conveting to adian meaue fit may be helpful in cetain poblem. Thi infomation will be ueful in Poblem c c 79. Atonomy. The un i about mi fom the Eath. If the angle ubtended by the diamete of the un on the uface of the Eath i ad, appoximately what i the diamete of the un to the neaet thouand mile in tandad decimal notation? 80. Atonomy. The moon i about 38,000 kilomete fom the Eath. If the angle ubtended by the diamete of the moon on the uface of the Eath i ad, appoximately what i the diamete of the moon to the neaet hunded kilomete? 8. Photogaphy. The angle of view of a 000-mm telephoto len i 2.5. At 750 ft, what i the width of the field of view to the neaet foot? 82. Photogaphy. The angle of view of a 300-mm len i 8. At 500 ft, what i the width of the field of view to the neaet foot? SECTION 5- Tigonometic Function Definition of the Tigonometic Function Calculato Evaluation of Tigonometic Function Definition of the Tigonometic Function Altenate Fom Exact Value fo Special Angle and Real Numbe Summay of Special Angle Value In thi ection we define tigonometic function with angle domain, whee angle can have eithe degee o adian meaue. We alo how how cicula function ae elated to tigonometic function o that you will be able to move eaily fom one to the othe, a needed. Definition of the Tigonometic Function We ae now eady to define tigonometic function with angle domain. Since we have aleady defined the cicula function with eal numbe domain, we can take advantage of thee eult and define the tigonometic function with angle domain in tem of the cicula function. To each of the ix cicula function we aociate a tigonometic function of the ame name. If i an angle, in eithe adian o degee meaue, we aign value to in, co, tan, cc, ec, and cot a given in Definition.