Section 53 Angles and Their Measure


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1 5 5 TRIGONOMETRIC FUNCTIONS Section 5 Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles, degee and adian. FIGURE Angle o angle PVQ o V. n Q Teminal V Initial P m Angles The study of tigonomety depends on the concept of angle. An angle is fomed by otating a half line, called a ay, aound its endpoint. One ay m, called the initial of the angle, emains fixed; a second ay n, called the teminal of the angle, stats in the initial position and otates aound the common endpoint V in a plane until it eaches its teminal position. The common endpoint V is called the vetex (see Fig. ). A counteclockwise otation poduces a positive angle, and a clockwise otation poduces a negative angle, as shown in Figues (a) and (b). The amount of otation in eithe diection is not esticted. Two diffeent angles may have the same initial and teminal s, as shown in Figue (c). Such angles ae said to be coteminal. FIGURE Angles and otation. Teminal Initial Teminal Teminal Initial negative positive Initial and coteminal (a) (b) (c) An angle in a ectangula coodinate system is said to be in standad position if its vetex is at the oigin and the initial is along the positive x axis. If the teminal of an angle in standad position lies along a coodinate axis, the angle is said to be a quadantal angle. If the teminal does not lie along a coodinate axis, then the angle is often efeed to in tems of the quadant in which the teminal lies (Fig. ). FIGURE Angles in standad positions. II III Teminal y Initial I IV x II III Teminal y Initial is a quadantal angle is a thidquadant angle is a secondquadant angle (a) (b) (c) I IV x II III Teminal y Initial I IV x
2 Degees and Radian Measue 5 Angles and Thei Measue 55 Just as line segments ae measued in centimetes, metes, inches, o miles, angles ae measued in diffeent units. The two most commonly used units fo angle measue ae degee and adian. DEFINITION DEGREE MEASURE An angle fomed by one complete otation is said to have a measue of 60 degees (60 ). An angle fomed by 60 of a complete otation is said to have a measue of degee ( ). The symbol denotes degees. Cetain angles have special names. Figue shows a staight angle, a ight angle, an acute angle, and an obtuse angle. FIGURE Types of angles Staight angle Right angle Acute angle Obtuse angle otation otation (0 90) (90 80) (a) (b) (c) (d) Two positive angles ae complementay if thei sum is 90 ; they ae supplementay if thei sum is 80. A degee can be divided futhe using decimal notation. Fo example,.75 epesents an angle of degee measue plus theequates of degee. A degee can also be divided futhe using minutes and seconds just as an hou is divided into minutes and seconds. Each degee is divided into 60 equal pats called minutes, and each minute is divided into 60 equal pats called seconds. Symbolically, minutes ae epesented by and seconds by. Thus, is a concise way of witing degees, minutes, and seconds. Decimal degees (DD) ae useful in some instances and degees minutes seconds (DMS) ae useful in othes. You should be able to go fom one fom to the othe as demonstated in Example. CONVERSION ACCURACY If an angle is measued to the neaest second, the conveted decimal fom should not go beyond thee decimal places, and vice vesa.
3 56 5 TRIGONOMETRIC FUNCTIONS EXAMPLE Fom DMS to DD and Back (A) Convet 7 to decimal degees. (B) Convet 05.8 to degee minute second fom. Solutions (A) , (B) (0.8.60) ( ) MATCHED PROBLEM (A) Convet 9 7 to DD fom. (B) Convet 7.65 to DMS fom. Some scientific and some gaphic calculatos can convet the DD and DMS foms automatically, but the pocess diffes significantly among the vaious types of calculatos. Check you owne s manual fo you paticula calculato. The convesion methods outlined in Example show you the easoning behind the pocess, and ae sometimes easie to use than the automatic methods fo some calculatos. Degee measue of angles is used extensively in engineeing, suveying, and navigation. Anothe unit of angle measue, called the adian, is bette suited fo cetain mathematical developments, scientific wok, and engineeing applications. DEFINITION RADIAN MEASURE If the vetex of an angle is placed at the cente of a cicle with adius 0, and the length of the ac opposite on the cicumfeence is s, then the adian measue of is given by Also, s adians s s O
4 5 Angles and Thei Measue 57 DEFINITION continued If s, then adian Thus, one adian is the size of the cental angle of a cicle that intecepts an ac the same length as the adius of the cicle. [Note: s and must be measued in the O adian same units. Also note that is being used in two ways: is the name of an angle and is the measue of the angle. The context detemines the choice. Thus, when we wite s/, we mean the adian measue of angle is s/.] s EXAMPLE Solution Computing Radian Measue What is the adian measue of a cental angle opposite an ac of metes in a cicle of adius 6 metes? s metes 6 metes adians MATCHED PROBLEM Remak What is the adian measue of a cental angle opposite an ac of 60 feet in a cicle of adius feet? Radian measue is a unitless numbe. The units in which the ac length and adius ae measued cancel; hence, we ae left with a unitless, o pue, numbe. Fo this eason, the wod adian is often omitted when we ae dealing with the adian measue of angles unless a special emphasis is desied. Exploe/Discuss Discuss why the adian measue of an angle is independent of the size of the cicle having the angle as a cental angle. Fom Degees to Radians and Vice Vesa What is the adian measue of an angle of 80? A cental angle of 80 is subtended by an ac of the cicumfeence of a cicle. Thus, if C is the cicumfeence of a cicle, then of the cicumfeence is given by s C and s ad
5 58 5 TRIGONOMETRIC FUNCTIONS Hence, 80 coesponds to * ad. This is impotant to emembe, since the adian measues of many special angles can be obtained fom this coespondence. Fo example, 90 is 80 /; theefoe, 90 coesponds to / ad. Exploe/Discuss Wite the adian measue of each of the following angles in the fom a a, whee a and b ae positive integes and faction is educed to b b lowest tems: 5, 0, 5, 60, 75, 90, 05, 0, 5, 50, 65, 80. Some key esults fom Exploe/Discuss ae summaized in Figue 5 fo easy efeence. These coespondences and multiples of them will be used extensively in wok that follows. FIGURE 5 Radian degee coespondences / 60 / 5 / 0 / / In geneal, the following popotion can be used to convet degee measue to adian measue and vice vesa. RADIAN DEGREE CONVERSION FORMULAS deg 80 ad ad Radians to degees deg 80 ad ad o ad ad 80 deg Degees to adians [Note: The popotion on the left is usually easie to emembe. Also we will omit units in calculations until the final answe. If you calculato does not have a key labeled, use.59.] *The constant has a long and inteesting histoy; a few impotant dates ae listed below: 650 B.C. Rhind Papyus 56 0 B.C. Achimedes ( ) A.D. 6 Liu Hui.59 A.D. 70 Tsu Ch ungchih A.D. 67 Leibniz ( ) (This and othe seies can be used to compute to any decimal accuacy desied.) A.D. 76 Johann Lambet Showed to be iational ( as a decimal is nonepeating and nonteminating)
6 5 Angles and Thei Measue 59 Some scientific and gaphic calculatos can automatically convet adian measue to degee measue, and vice vesa. Check the owne s manual fo you paticula calculato. EXAMPLE Radian Degee Convesions (A) Find the adian measue, exact and to thee significant digits, of an angle of 75. (B) Find the degee measue, exact and to fou significant digits, of an angle of 5 adians. (C) Find the adian measue to two decimal places of an angle of. Solutions Exact Thee significant digits ad 5 (A) ad. 80 deg (75) 80 FIGURE 6 Automatic convesion. Exact Fou significant digits (B) 900 deg 80 ad ad 80 (5) 86.5 (C) 60. ad ad 80 deg (.) Change to DD fist. To two decimal places Figue 6 shows the thee peceding convesions done automatically on a gaphing calculato. MATCHED PROBLEM (A) Find the adian measue, exact and to thee significant digits, of an angle of 0. (B) Find the degee measue, exact and to thee significant digits, of an angle of adian. (C) Find the adian measue to thee decimal places of an angle of 5. Remak We will wite in place of deg and ad when it is clea fom the context whethe we ae dealing with degee o adian measue. EXAMPLE Engineeing A belt connects a pulley of inch adius with a pulley of 5inch adius. If the lage pulley tuns though 0 adians, though how many adians will the smalle pulley tun?
7 60 5 TRIGONOMETRIC FUNCTIONS Solution Fist we daw a sketch (Fig. 7). FIGURE 7 P 5 in. Q in. When the lage pulley tuns though 0 adians, the point P on its cicumfeence will tavel the same distance (ac length) that point Q on the smalle cicle tavels. Fo the lage pulley, s s (5)(0) 50 inches Fo the smalle pulley, s 50 5 adians MATCHED PROBLEM In Example, though how many adians will the lage pulley tun if the smalle pulley tuns though adians? Answes to Matched Poblems 80. (A) 9.9 (B) ad. (A).9 (B) 57. (C) ad EXERCISE 5 In all poblems, if angle measue is expessed by a numbe that is not in degees, it is assumed to be in adians. A Find the degee measue of each of the angles in Poblems 6, keeping in mind that an angle of one complete otation coesponds to otation. 5 otation. otation. 8 otation 5. 8 otations 6. Find the adian measue of a cental angle opposite an ac s in a cicle of adius, whee and s ae as given in Poblems centimetes, s centimetes 8. 8 inches, s 6 inches 9. feet, s 0 feet 0. 8 metes, s 7 metes otations
8 5 Angles and Thei Measue 6 Find the adian measue of each angle in Poblems 6, keeping in mind that an angle of one complete otation coesponds to adians.. 8 otation. 6 otation. 5. otation 5. otations 6. B Find the exact adian measue, in tems of, of each angle in Poblems , 60, 90, 0, 50, , 0, 80, 0, 00, , 90, 5, , 80, 70, 60 Find the exact degee measue of each angle in Poblems...,,,, 5,..,,,,,, Convet each angle in Poblems 5 8 to decimal degees to thee decimal places Convet each angle in Poblems 9 to degee minute second fom ,,,, 5 6, Find the adian measue to thee decimal places fo each angle in Poblems Find the degee measue to two decimal places fo each angle in Poblems otation 8 otations Indicate whethe each angle in Poblems 5 6 is a I, II, III, o IV quadant angle o a quadantal angle. All angles ae in standad position in a ectangula coodinate system. (A sketch may be of help in some poblems.) Vebally descibe the meaning of a cental angle in a cicle with adian measue. 66. Vebally descibe the meaning of an angle with degee measue. C Which angles in Poblems 67 7 ae coteminal with 0 if all angles ae placed in standad position in a ectangula coodinate system? Which angles in poblems 7 78 ae coteminal with / if all angles ae placed in standad position in a ectangula coodinate system? APPLICATIONS 79. Cicumfeence of the Eath. The ealy Geeks used the popotion s/c /60, whee s is an ac length on a cicle, is degee measue of the coesponding cental angle, and C is the cicumfeence of the cicle (C ). Eatosthenes (0 B.C.), in his famous calculation of the cicumfeence of the eath, easoned as follows: He knew at Syene (now Aswan) duing the summe solstice the noon sun was diectly ovehead and shined on the wate 6
9 6 5 TRIGONOMETRIC FUNCTIONS staight down a deep well. On the same day at the same time, 5,000 stadia (appox. 500 mi) due noth in Alexandia, sun ays cossed a vetical pole at an angle of 7.5 as indicated in the figue. Cay out Eatosthenes calculation fo the cicumfeence of the eath to the neaest thousand miles. (The cuent calculation fo the equatoial cicumfeence is,90 mi.) Eath Alexandia 7.5 Syene Well Sun ays 80. Cicumfeence of the Eath. Repeat Poblem 79 with the sun cossing the vetical pole in Alexandia at Cicumfeence of the Eath. In Poblem 79, vebally explain how in the figue was detemined. 8. Cicumfeence of the Eath. Vebally explain how the adius, suface aea, and volume of the eath can be detemined fom the esult of Poblem Radian Measue. What is the adian measue of the lage angle made by the hands of a clock at :0? Expess the answe exactly in tems of. 8. Radian Measue. What is the adian measue of the smalle angle made by the hands of a clock at :0? Expess the answe exactly in tems of. 85. Engineeing. Though how many adians does a pulley of 0centimete diamete tun when 0 metes of ope ae pulled though it without slippage? 86. Engineeing. Though how many adians does a pulley of 6inch diamete tun when feet of ope ae pulled though it without slippage? 87. Astonomy. A line fom the sun to the Eath sweeps out an angle of how many adians in week? Assume the Eath s obit is cicula and thee ae 5 weeks in a yea. Expess the answe in tems of and as a decimal to two decimal places. 88. Astonomy. A line fom the cente of the Eath to the equato sweeps out an angle of how many adians in 9 hous? Expess the answe in tems of and as a decimal to two decimal places. 89. Engineeing. A tail bike has a font wheel with a diamete of 0 centimetes and a back wheel of diamete 60 centimetes. Though what angle in adians does the font wheel tun if the back wheel tuns though 8 adians? 90. Engineeing. In Poblem 89, though what angle in adians will the back wheel tun if the font wheel tuns though 5 adians? The ac length on a cicle is easy to compute if the coesponding cental angle is given in adians and the adius of the cicle is known (s ). If the adius of a cicle is lage and a cental angle is small, then an ac length is often used to appoximate the length of the coesponding chod as shown in the figue. If an angle is given in degee measue, conveting to adian measue fist may be helpful in cetain poblems. This infomation will be useful in Poblems 9 9. c s 9. Astonomy. The sun is about mi fom the eath. If the angle subtended by the diamete of the sun on the suface of the eath is 9. 0 ad, appoximately what is the diamete of the sun to the neaest thousand miles in standad decimal notation? 9. Astonomy. The moon is about 8,000 km fom the eath. If the angle subtended by the diamete of the moon on the suface of the eath is ad, appoximately what is the diamete of the moon to the neaest hunded kilometes? 9. Photogaphy. The angle of view of a,000mm telephoto lens is.5. At 750 ft, what is the width of the field of view to the neaest foot? 9. Photogaphy. The angle of view of a 00mm lens is 8. At 500 ft, what is the width of the field of view to the neaest foot? c s
SECTION 53 Angles and Their Measure
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